aarjaneiro/planetmath huggingface dataset:
derivation of the generating series for the Stirling numbers of the second kind

derivation of the generating series for the Stirling numbers of the second kind

The

derivation

of the generating series is much simpler if one makes use of the

composition

lemma for exponential generating series. We are looking for the generating series for sets of

nonempty sets

, so in the notation of Jackson and Goulden, we have the set

decomposition

:

𝒜

⁢

⟶

~

⁢

𝒰

⊛

(

𝒰

\

{

∅

}

)

where

𝒰

is the set of all canonical unordered sets,

𝒜

is the set which we are interested in counting, and

⊛

is star-composition of sets of labelled combinatorial objects.

The set

𝒰

has one object in it of each weight, and so has exponential generating series:

[

(

𝒰

,

ω

)

]

e

⁢

(

x

)

=

∑

n

≥

0

x

n

n

!

=

e

x

The set

𝒰

\

{

∅

}

then has generating series:

[

(

𝒰

\

{

∅

}

,

ω

)

]

e

⁢

(

x

)

=

e

x

-

1

So, by the star composition lemma and the above decomposition,

[

(

𝒜

,

ω

)

]

e

⁢

(

x

)

=

[

(

𝒰

⊛

(

𝒰

\

{

∅

}

)

,

ω

)

]

e

⁢

(

x

)

=

(

[

(

𝒰

,

ω

)

]

e

∘

[

(

𝒰

\

{

∅

}

,

ω

)

]

e

)

⁢

(

x

)

=

e

e

x

-

1

By tensoring the weight function

ω

with a weight function

λ

counting the number of parts each

set partition

contains, we get

[

(

𝒜

,

ω

⊗

λ

)

]

e

,

o

⁢

(

x

,

t

)

=

e

t

⁢

(

e

x

-

1

)

using a derivation

similar

to the one above.

Title

derivation of the generating series for the

Stirling numbers

of the second kind

Canonical name

DerivationOfTheGeneratingSeriesForTheStirlingNumbersOfTheSecondKind

Date of creation

2013-03-22 15:13:35

Last modified on

2013-03-22 15:13:35

Owner

cgibbard (959)

Last modified by

cgibbard (959)

Numerical id

6

Author

cgibbard (959)

Entry type

Derivation

Classification

msc 05A15

Generated on Thu Feb 8 20:37:05 2018 by

LaTeXML
Stirling polynomial

Stirling polynomial

Stirling’s

polynomials

S

k

⁢

(

x

)

are defined by the

generating function

(

t

1

-

e

-

t

)

x

+

1

=

∑

k

=

0

S

k

⁢

(

x

)

k

!

⁢

t

k

.

The sequence

S

k

⁢

(

x

-

1

)

is of binomial type, since

S

k

⁢

(

x

+

y

-

1

)

=

∑

i

=

0

k

(

k

i

)

⁢

S

i

⁢

(

x

-

1

)

⁢

S

k

-

i

⁢

(

y

-

1

)

. Moreover, this basic recursion holds:

S

k

⁢

(

x

)

=

(

x

-

k

)

⁢

S

k

⁢

(

x

-

1

)

x

+

k

⁢

S

k

-

1

⁢

(

x

+

1

)

.

These are the first polynomials:

1.

S

0

⁢

(

x

)

=

1

;

2.

S

1

⁢

(

x

)

=

1

2

⁢

(

x

+

1

)

;

3.

S

2

⁢

(

x

)

=

1

12

⁢

(

3

⁢

x

2

+

5

⁢

x

+

2

)

;

4.

S

3

⁢

(

x

)

=

1

8

⁢

(

x

3

+

2

⁢

x

2

+

x

)

;

5.

S

4

⁢

(

x

)

=

1

240

⁢

(

15

⁢

x

4

+

30

⁢

x

3

+

5

⁢

x

2

-

18

⁢

x

-

8

)

.

In addition we have these special values:

1.

S

k

⁢

(

-

m

)

=

(

-

1

)

k

(

k

+

m

-

1

k

)

⁢

S

k

+

m

-

1

,

m

-

1

, where

S

m

,

n

denotes

Stirling numbers

of the

second kind

. Conversely,

S

n

,

m

=

(

-

1

)

n

-

m

⁢

(

n

m

)

⁢

S

n

-

m

⁢

(

-

m

-

1

)

;

2.

S

k

⁢

(

-

1

)

=

δ

k

,

0

;

3.

S

k

⁢

(

0

)

=

(

-

1

)

k

⁢

B

k

, where

B

k

are Bernoulli’s numbers;

4.

S

k

⁢

(

1

)

=

(

-

1

)

k

+

1

⁢

(

(

k

-

1

)

⁢

B

k

+

k

⁢

B

k

-

1

)

;

5.

S

k

⁢

(

2

)

=

(

-

1

)

k

2

⁢

(

(

k

-

1

)

⁢

(

k

-

2

)

⁢

B

k

+

3

⁢

k

⁢

(

k

-

2

)

⁢

B

k

-

1

+

2

⁢

k

⁢

(

k

-

1

)

⁢

B

k

-

2

)

;

6.

S

k

⁢

(

k

)

=

k

!

;

7.

S

k

⁢

(

m

)

=

(

-

1

)

k

(

m

k

)

⁢

s

m

+

1

,

m

+

1

-

k

, where

s

m

,

n

are Stirling numbers of the

first kind

. They may be recovered by

s

n

,

m

=

(

-

1

)

n

-

m

⁢

(

n

-

1

n

-

m

)

⁢

S

n

-

m

⁢

(

n

-

1

)

.

Explicit

representations

involving Stirling numbers can be deduced with Lagrange’s interpolation formula:

S

k

⁢

(

x

)

=

∑

n

=

0

k

(

-

1

)

k

-

n

⁢

S

k

+

n

,

n

⁢

(

x

+

n

n

)

⁢

(

x

+

k

+

1

k

-

n

)

(

k

+

n

n

)

=

∑

n

=

0

k

(

-

1

)

n

⁢

s

k

+

n

+

1

,

n

+

1

⁢

(

x

-

k

n

)

⁢

(

x

-

k

-

n

-

1

k

-

n

)

(

k

+

n

k

)

.

These following formulae hold as well:

(

k

+

m

k

)

⁢

S

k

⁢

(

x

-

m

)

=

∑

i

=

0

k

(

-

1

)

k

-

i

⁢

(

k

+

m

i

)

⁢

S

k

-

i

+

m

,

m

⁢

S

i

⁢

(

x

)

,

(

k

-

m

k

)

⁢

S

k

⁢

(

x

+

m

)

=

∑

i

=

0

k

(

k

-

m

i

)

⁢

s

m

,

m

-

k

+

i

⁢

S

i

⁢

(

x

)

.

Title

Stirling polynomial

Canonical name

StirlingPolynomial

Date of creation

2013-03-22 15:38:36

Last modified on

2013-03-22 15:38:36

Owner

kronos (12218)

Last modified by

kronos (12218)

Numerical id

9

Author

kronos (12218)

Entry type

Definition

Classification

msc 05A15

Generated on Thu Feb 8 20:37:43 2018 by

LaTeXML
Wedderburn-Etherington number

Wedderburn-Etherington number

The

n

th

Wedderburn-Etherington number

counts how many

weakly binary trees

can be constructed such that each

graph vertex

(not counting the

root vertex

) is

adjacent

to no more than three other such vertices, for a given number

n

of nodes. The first few Wedderburn-Etherington numbers are 1, 1, 1, 2, 3, 6, 11, 23, 46, 98, 207, 451, 983, etc. listed in A001190 of Sloane’s OEIS. Michael Somos gives the following

recurrence relations

:

a

2

⁢

n

=

1

2

⁢

a

n

⁢

a

n

+

1

+

∑

i

=

1

n

a

i

⁢

a

2

⁢

n

-

i

and

a

2

⁢

n

-

1

=

∑

i

=

0

n

-

1

a

i

+

1

⁢

a

2

⁢

n

-

i

with

a

1

=

a

2

=

1

in both

relations

.

Title

Wedderburn-Etherington number

Canonical name

WedderburnEtheringtonNumber

Date of creation

2013-03-22 16:49:32

Last modified on

2013-03-22 16:49:32

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 05A15

Synonym

Wedderburn Etherington number

Synonym

Etherington-Wedderburn number

Generated on Thu Feb 8 20:37:45 2018 by

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Takeuchi function

Takeuchi function

The

Takeuchi function

is a triply

recursive

3-parameter

function

originally defined by Ichiro Takeuchi in 1978 as

t

⁢

(

x

,

y

,

z

)

=

y

if

x

≤

y

and

t

⁢

(

x

,

y

,

z

)

=

t

⁢

(

t

⁢

(

x

-

1

,

y

,

z

)

,

t

⁢

(

y

-

1

,

z

,

x

)

,

t

⁢

(

z

-

1

,

x

,

y

)

)

otherwise. Later John McCarthy simplified the definition of the function as

t

⁢

(

x

,

y

,

z

)

=

y

if

x

≤

y

,

t

⁢

(

x

,

y

,

z

)

=

z

if

y

≤

z

and

t

⁢

(

x

,

y

,

z

)

=

x

in all other cases.

For example,

t

⁢

(

194

,

13

,

5

)

=

194

since 194 is not less than 13, and 13 is not less than 5. The return value of the function “is on no practical significance,” but the function itself “is useful for benchmark testing of programming languages.” (Finch, 2003) The function

T

⁢

(

x

,

y

,

z

)

is the number of times

t

calls itself to obtain the return value. A properly optimized implementation of the function in a given programming language should not require more

recursion

than

T

indicates.

References

1

Steven R. Finch

Mathematical Constants

New York: Cambridge University Press (2003): 321

Title

Takeuchi function

Canonical name

TakeuchiFunction

Date of creation

2013-03-22 17:33:07

Last modified on

2013-03-22 17:33:07

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 05A16

Generated on Thu Feb 8 20:37:47 2018 by

LaTeXML
Takeuchi number

Takeuchi number

The

n

th

Takeuchi number

T

n

is the value of the

function

T

⁢

(

n

,

0

,

n

+

1

)

which measures how many times the

Takeuchi function

t

⁢

(

x

,

y

,

z

)

has to call itself to give the answer starting with

x

=

n

,

y

=

0

,

z

=

n

+

1

. For example, the second Takeuchi number is 4, since

t

⁢

(

2

,

0

,

3

)

requires four

recursions

to obtain the answer 2. The first few Takeuchi numbers are 0, 1, 4, 14, 53, 223, 1034, 5221, 28437, listed in A000651 of Sloane’s OEIS. Prellberg gives a

formula

for the asymptotic growth of the Takeuchi numbers:

T

n

⁢

c

⁢

B

n

⁢

exp

⁡

(

1

2

⁢

W

⁢

(

n

)

2

)

, where

c

is the Takeuchi-Prellberg constant (approximately 2.2394331),

B

n

is the

n

th

Bernoulli number

and

W

⁢

(

x

)

is Lambert’s

W

function.

References

1

Steven R. Finch

Mathematical Constants

New York: Cambridge University Press (2003): 321

2

T. Prellberg, “On the asymptotics of Takeuchi numbers”,

Symbolic computation

,

number theory

,

special functions

, physics and

combinatorics

, Dordrecht: Kluwer Acad. Publ. (2001): 231 - 242.

Title

Takeuchi number

Canonical name

TakeuchiNumber

Date of creation

2013-03-22 17:33:09

Last modified on

2013-03-22 17:33:09

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 05A16

Generated on Thu Feb 8 20:37:49 2018 by

LaTeXML
partition function

partition function

The

partition function

p

⁢

(

n

)

is defined to be the number of

partitions

of the

integer

n

. The

sequence

of values

p

⁢

(

0

)

,

p

⁢

(

1

)

,

p

⁢

(

2

)

,

…

is Sloane’s A000041 and begins

1

,

1

,

2

,

3

,

5

,

7

,

11

,

15

,

22

,

30

,

…

. This

function

grows very quickly, as we see in the following

theorem

due to Hardy and Ramanujan (

http://planetmath.org/SrinivasaRamanujan

).

Theorem 1

As

n

→

∞

, the ratio of

p

⁢

(

n

)

and

e

π

⁢

2

⁢

n

/

3

4

⁢

n

⁢

3

approaches 1.

The

generating function

of

p

⁢

(

n

)

is called

F

: by definition

F

⁢

(

x

)

=

∑

n

=

0

∞

p

⁢

(

n

)

⁢

x

n

.

F

can be written as an infinite product:

F

⁢

(

x

)

=

∏

i

=

1

∞

(

1

-

x

i

)

-

1

.

To see this, expand each term in the

product

as a

power series

:

∏

i

=

1

∞

(

1

+

x

i

+

x

2

⁢

i

+

x

3

⁢

i

+

⋯

)

.

Now expand this as a power series. Given a partition of

n

with

a

i

parts of size

i

≥

1

, we get a term

x

n

in this expansion by choosing

x

a

1

from the first term in the product,

x

2

⁢

a

2

from the second,

x

3

⁢

a

3

from the third and so on. Clearly any term

x

n

in the expansion arises in this way from a partition of

n

.

One can prove in the same way that the generating function

F

m

for the number

p

m

⁢

(

n

)

of partitions of

n

into at most

m

parts (or equivalently into parts of size at most

m

) is

F

m

⁢

(

x

)

=

∏

i

=

1

m

(

1

-

x

i

)

-

1

.

References

1

G. H. Hardy and E. M. Wright,

An

Introduction

to the Theory of Numbers

, Oxford University Press, 2003.

Title

partition function

Canonical name

PartitionFunction

Date of creation

2013-03-22 15:57:55

Last modified on

2013-03-22 15:57:55

Owner

silverfish (6603)

Last modified by

silverfish (6603)

Numerical id

9

Author

silverfish (6603)

Entry type

Definition

Classification

msc 05A17

Related topic

IntegerPartition

Related topic

NonMultiplicativeFunction

Defines

partition generating function

Generated on Thu Feb 8 20:38:07 2018 by

LaTeXML
prime partition

prime partition

A

prime partition

is a

partition

(

http://planetmath.org/IntegerPartition

) of a given

positive integer

n

consisting only of

prime numbers

. For example, a prime partition of 42 is 29 + 5 + 5 + 3.

If we accept partitions of length 1 as valid partitions, then it is

obvious

that only prime numbers have prime partitions of length 1. Not accepting 1 as a prime number makes the problem of prime partitions more interesting, otherwise there would always be for a given

n

, if nothing else, a prime partition consisting of

n

1s. Almost as bad, however, is a partion of

n

into

⌊

n

2

⌋

2s and 3s.

Both Goldbach’s

conjecture

and Levy’s conjecture can be restated in terms of prime partitions thus: for any

even integer

n

>

2

there is always a prime partition of length 2, and for any odd integer

n

>

5

there is always a prime partition of length 3 with at most 2 distinct

elements

.

Assuming Goldbach’s conjecture is true, the most efficient prime partition of an even integer is of length 2, while Vinogradov’s

theorem

has proven the most efficient prime partition of a sufficiently large

composite

odd integer is of length 3.

Title

prime partition

Canonical

name

PrimePartition

Date of creation

2013-03-22 17:28:02

Last modified on

2013-03-22 17:28:02

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 05A17

Classification

msc 11P99

Generated on Thu Feb 8 20:38:09 2018 by

LaTeXML
arrows relation

arrows relation

Let

[

X

]

α

=

{

Y

⊆

X

∣

|

Y

|

=

α

}

, that is, the set of subsets of

X

of size

α

. Then given some

cardinals

κ

,

λ

,

α

and

β

κ

→

(

λ

)

β

α

states that for any set

X

of size

κ

and any

function

f

:

[

X

]

α

→

β

, there is some

Y

⊆

X

and some

γ

∈

β

such that

|

Y

|

=

λ

and for any

y

∈

[

Y

]

α

,

f

⁢

(

y

)

=

γ

.

In words, if

f

is a

partition

of

[

X

]

α

into

β

subsets then

f

is

constant

on a subset of size

λ

(a

homogeneous

subset).

As an example, the

pigeonhole principle

is the statement that if

n

is finite and

k

<

n

then:

n

→

2

k

1

That is, if you try to partition

n

into fewer than

n

pieces then one piece has more than one

element

.

Observe that if

κ

→

(

λ

)

β

α

then the same statement holds if:

•

κ

is made larger (since the

restriction

of

f

to a set of size

κ

can be considered)

•

λ

is made smaller (since a subset of the homogeneous set will suffice)

•

β

is made smaller (since any partition into fewer than

β

pieces can be expanded by adding

empty sets

to the partition)

•

α

is made smaller (since a partition

f

of

[

κ

]

γ

where

γ

<

α

can be extended to a partition

f

′

of

[

κ

]

α

by

f

′

⁢

(

X

)

=

f

⁢

(

X

γ

)

where

X

γ

is the

γ

smallest elements of

X

)

κ

↛

(

λ

)

β

α

is used to state that the corresponding

→

relation

is false.

References

•

Jech, T.

Set Theory

, Springer-Verlag, 2003

•

Just, W. and Weese, M.

Topics in Discovering Modern Set Theory, II

,

American Mathematical Society

, 1996

Title

arrows relation

Canonical name

ArrowsRelation

Date of creation

2013-03-22 17:48:54

Last modified on

2013-03-22 17:48:54

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

5

Author

Henry (455)

Entry type

Definition

Classification

msc 05A18

Classification

msc 03E05

Related topic

PartitionsLessThanCofinality

Related topic

ErdosRadoTheorem

Defines

homogeneous

Defines

arrows

Defines

homogeneous set

Defines

homogeneous subset

Generated on Thu Feb 8 20:38:11 2018 by

LaTeXML
Bell number

Bell number

The

Bell number

, denoted

B

⁢

(

n

)

is the total number of

partitions

of a set with

n

elements. For

n

=

0

, we have

B

⁢

(

0

)

=

1

. For

n

≥

1

, we have

B

⁢

(

n

)

=

∑

k

=

0

n

S

⁢

(

n

,

k

)

for

⁢

n

≥

1

where

S

⁢

(

n

,

k

)

are the

Stirling numbers

of the

second kind

.

Proposition 1

.

B

⁢

(

n

+

1

)

=

∑

k

=

0

n

(

n

k

)

⁢

B

⁢

(

k

)

Proof.

We count the number of partitions of a set of

n

+

1

elements, depending on the size of the block containing the

n

+

1

st element. If the block has size

j

for

1

≤

j

≤

n

+

1

then we have

(

n

j

-

1

)

choices for the

j

-

1

other

elements of

the block. The remaining

n

+

1

-

j

elements can be partitioned in

B

⁢

(

n

+

1

-

j

)

ways. We have therefore that:

B

⁢

(

n

+

1

)

=

∑

j

=

1

n

+

1

(

n

j

-

1

)

⁢

B

⁢

(

n

+

1

-

j

)

=

∑

j

=

1

n

+

1

(

n

n

+

1

-

j

)

⁢

B

⁢

(

n

+

1

-

j

)

=

∑

k

=

0

n

(

n

k

)

⁢

B

⁢

(

k

)

∎

Using the formula above, one can easily derive the first few Bell numbers. Starting with

n

=

0

, the first ten Bell numbers are 1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147.

Title

Bell number

Canonical name

BellNumber

Date of creation

2013-03-22 14:47:07

Last modified on

2013-03-22 14:47:07

Owner

aoh45 (5079)

Last modified by

aoh45 (5079)

Numerical id

7

Author

aoh45 (5079)

Entry type

Definition

Classification

msc 05A18

Classification

msc 11B73

Related topic

StirlingNumbersSecondKind

Generated on Thu Feb 8 20:38:14 2018 by

LaTeXML
Bell’s triangle

Bell’s triangle

Bell’s triangle

or

Aitken’s array

or

Peirce triangle

is a triangular

arrangement

of

integers

in which the top row has a single 1, and each subsequent row begins with the last number of the previous row, and each remaining number in a row is the sum of the number to the left to the number above left.

The first eight rows are:

1

1

2

2

3

5

5

7

10

15

15

20

27

37

52

52

67

87

114

151

203

203

255

322

409

523

674

877

877

1080

1335

1657

2066

2589

3263

4140

⋮

⋮

⋮

These are listed in A011971 of Sloane’s OEIS.

The left outermost diagonal gives the

Bell numbers

(1, 2, 5, 15, 52, 203, 877, 4140, etc.) in order.

Title

Bell’s triangle

Canonical name

BellsTriangle

Date of creation

2013-03-22 16:49:16

Last modified on

2013-03-22 16:49:16

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Data Structure

Classification

msc 05A18

Classification

msc 11B73

Synonym

Bell triangle

Synonym

Aitken’s array

Synonym

Aitken array

Synonym

Peirce triangle

Synonym

Peirce’s triangle

Generated on Thu Feb 8 20:38:16 2018 by

LaTeXML
expectation example

expectation example

The

contraharmonic mean

of several positive numbers

u

1

,

u

2

,

…

,

u

n

is defined as

c

:=

u

1

2

+

u

2

2

+

…

+

u

n

2

u

1

+

u

2

+

…

+

u

n

.

This has certain applications; one
of them is by [1] the following.

If

⟨

u

1

,

u

2

,

…

,

u

n

⟩

is the

distribution

of the seats of

n

parties,

s

the total number of seats in the body of delegates
(

u

1

+

u

2

+

…

+

u

n

=

s

),
and one draws a random seat (with

probability

1

/

s

), then the size of the drawn
delegate’s party has the

expected value

u

1

s

⋅

u

1

+

u

2

s

⋅

u

2

+

…

+

u

n

s

⋅

u

n

=

c

.

References

1

Caulier, Jean-François

: The

interpretation

of the Laakso–Taagepera effective number of parties.  –

Documents de travail du Centre d’Economie de la Sorbonne

(2011.06).

Title

expectation example

Canonical name

ExpectationExample

Date of creation

2013-11-05 17:59:13

Last modified on

2013-11-05 17:59:13

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

2

Author

pahio (2872)

Entry type

Example

Classification

msc 05A18

Classification

msc 26E60

Classification

msc 60A05

Generated on Thu Feb 8 20:38:18 2018 by

LaTeXML
generating function for the reciprocal Catalan numbers

generating function for the reciprocal Catalan numbers

The series

1

+

x

+

x

2

2

+

x

3

5

+

x

4

14

+

x

5

42

+

x

6

132

+

x

7

429

+

⋯

whose coefficients are the reciprocal of the

Catalan numbers

(

2

⁢

n

n

)

n

+

1

, has as a

generating function

2

⁢

(

4

-

x

⁢

(

8

+

x

)

+

12

⁢

x

⁢

arctan

⁡

(

x

4

-

x

)

)

(

4

-

x

)

5

To deduce such a formula the easy way, one starts from the generating function of the reciprocal

central binomial coefficients

and having into account the relation

d

d

⁢

x

⁢

(

x

n

+

1

(

2

⁢

n

n

)

)

=

(

n

+

1

)

⁢

x

n

(

2

⁢

n

n

)

for each term in the corresponding series and applied to the

function

in the region of

uniform convergence

.
Another method is almost exaclty the same like in the derivation of the generating function for the reciprocal central binomial coefficients.

Title

generating function for the reciprocal Catalan numbers

Canonical name

GeneratingFunctionForTheReciprocalCatalanNumbers

Date of creation

2013-03-22 19:05:12

Last modified on

2013-03-22 19:05:12

Owner

juanman (12619)

Last modified by

juanman (12619)

Numerical id

10

Author

juanman (12619)

Entry type

Derivation

Classification

msc 05A19

Classification

msc 05A15

Classification

msc 05A10

Related topic

CatalanNumbers

Generated on Thu Feb 8 20:38:21 2018 by

LaTeXML
generating function for the reciprocal central binomial coefficients

generating function for the reciprocal central binomial coefficients

It is well known that the

sequence

called

central binomial coefficients

is defined by

(

2

⁢

n

n

)

and whose initial terms are

1

,

2

,

6

,

20

,

70

,

252

,

…

has a

generating function

1

1

-

4

⁢

x

.
But it is less known the fact that
the function

4

⁢

(

4

-

x

+

x

⁢

arcsin

⁡

(

x

2

)

)

(

4

-

x

)

3

has ordinary

power series

1

+

x

2

+

x

2

6

+

x

3

20

+

x

4

70

+

x

5

252

+

…

This means that such a function is a generating function for the

reciprocals

(

2

⁢

n

n

)

-

1

.

From that

expression

we can see that the numerical series

∑

n

=

0

∞

(

2

⁢

n

n

)

-

1

sums

4

⁢

(

3

+

π

6

)

3

⁢

3

which has the approximate value

1

,

7363998587187151

.

Reference:

1) Renzo Sprugnoli,

Sum of reciprocals of the Central Binomial Coefficients

,

Integers

: electronic
journal of combinatorial number theory, 6 (2006) A27, 1-18

Title

generating function for the reciprocal central binomial coefficients

Canonical

name

GeneratingFunctionForTheReciprocalCentralBinomialCoefficients

Date of creation

2013-03-22 18:58:09

Last modified on

2013-03-22 18:58:09

Owner

juanman (12619)

Last modified by

juanman (12619)

Numerical id

12

Author

juanman (12619)

Entry type

Result

Classification

msc 05A19

Classification

msc 11B65

Classification

msc 05A10

Classification

msc 05A15

Synonym

convergent series

Generated on Thu Feb 8 20:38:23 2018 by

LaTeXML
König-Egervary theorem

König-Egervary theorem

The

König-Egervary

theorem

states that in a finite matrix of 0’s and 1’s, the maximum numbers of 1’s such that no two are in a line, equals the minimum number of lines which collectively contain all the 1’s. Here line means row or column.

Take this matrix, for example,

[

1

0

0

0

0

1

1

0

0

0

0

1

0

1

0

0

0

1

0

1

0

0

0

1

1

0

1

1

1

0

]

Here the max and min numbers (always equal) are 4.

References

1

A. Chandra Babu, P. V. Ramakrishnan, “New Proofs of

Konig-Egervary Theorem

And Maximal Flow-Minimal Cut Capacity Theorem Using O. R. Techniques”

Indian J. Pure
Appl. Math.

22

(11) (1991): 905 - 911

Title

König-Egervary theorem

Canonical

name

KonigEgervaryTheorem

Date of creation

2013-03-22 16:33:47

Last modified on

2013-03-22 16:33:47

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 05A19

Synonym

Konig-Egervary theorem

Generated on Thu Feb 8 20:38:25 2018 by

LaTeXML
Pascal’s rule

Pascal’s rule

Pascal’s rule is the

binomial identity

(

n

k

)

+

(

n

k

-

1

)

=

(

n

+

1

k

)

where

1

≤

k

≤

n

and

(

n

k

)

is the

binomial coefficient

.

Title

Pascal’s rule

Canonical name

PascalsRule

Date of creation

2013-03-22 11:46:44

Last modified on

2013-03-22 11:46:44

Owner

KimJ (5)

Last modified by

KimJ (5)

Numerical id

10

Author

KimJ (5)

Entry type

Theorem

Classification

msc 05A19

Related topic

BinomialCoefficient

Related topic

VandermondeIdentity

Related topic

PascalsTriangle

Related topic

TheoremOfThePrimalRay

Related topic

Mm2

Related topic

Mm

Related topic

LeTheoremeDuRayonPrimal

Related topic

LeDeuxiemeTheoremeDuRayonPrimal

Related topic

AProofOfGoldbachConjecture

Related topic

AProofOfDePolignacConjectures

Related topic

FermatGhanouchiSeriesAmazingFermatGhanouchiSequ

Generated on Thu Feb 8 20:38:27 2018 by

LaTeXML