Little Johnny has recently learned about set theory. Now he is studying binary relations. You've probably heard the term "equivalence relation". These relations are very important in many areas of mathematics. For example, the equality of the two numbers is an equivalence relation.

A set ρ of pairs (a, b) of elements of some set A is called a binary relation on set A. For two elements a and b of the set A we say that they are in relation ρ, if pair $(a, b) \in \rho$, in this case we use a notation $a \stackrel{\rho}{\sim} b$.

Binary relation is equivalence relation, if: It is reflexive (for any a it is true that $a \stackrel{\rho}{\sim} a$); It is symmetric (for any a, b it is true that if $a \stackrel{\rho}{\sim} b$, then $b \stackrel{\rho}{\sim} a$); It is transitive (if $a \stackrel{\rho}{\sim} b$ and $b \stackrel{\rho}{\sim} c$, than $a \stackrel{\rho}{\sim} c$).

Little Johnny is not completely a fool and he noticed that the first condition is not necessary! Here is his "proof":

Take any two elements, a and b. If $a \stackrel{\rho}{\sim} b$, then $b \stackrel{\rho}{\sim} a$ (according to property (2)), which means $a \stackrel{\rho}{\sim} a$ (according to property (3)).

It's very simple, isn't it? However, you noticed that Johnny's "proof" is wrong, and decided to show him a lot of examples that prove him wrong.

Here's your task: count the number of binary relations over a set of size n such that they are symmetric, transitive, but not an equivalence relations (i.e. they are not reflexive).

Since their number may be very large (not 0, according to Little Johnny), print the remainder of integer division of this number by 10^9 + 7.


-----Input-----

A single line contains a single integer n (1 ≤ n ≤ 4000).


-----Output-----

In a single line print the answer to the problem modulo 10^9 + 7.


-----Examples-----
Input
1

Output
1

Input
2

Output
3

Input
3

Output
10



-----Note-----

If n = 1 there is only one such relation — an empty one, i.e. $\rho = \varnothing$. In other words, for a single element x of set A the following is hold: [Image].

If n = 2 there are three such relations. Let's assume that set A consists of two elements, x and y. Then the valid relations are $\rho = \varnothing$, ρ = {(x, x)}, ρ = {(y, y)}. It is easy to see that the three listed binary relations are symmetric and transitive relations, but they are not equivalence relations.

Difficulty: HARD
Tags: Dynamic programming

Combinatorics

Mathematics
Skill types: Dynamic programming

Starter code:

Solutions:
def main():
	mod = 10 ** 9 + 7
	n = int(input())
	a = [[0] * (n + 1) for i in range(n + 1)]
	a[0][0] = 1
	for i in range(1, n + 1):
		a[i][0] = a[i - 1][i - 1]
		for j in range(1, i + 1):
			a[i][j] = (a[i][j - 1] + a[i - 1][j - 1]) % mod
	print(a[n][n - 1])
main()


mod = 10 ** 9 + 7
n = int(input())
f = [0] * (n + 1)
f[0] = 1
count = 1
for i in range(1, n + 1):
	for j in range(0, i):
		f[(count + j + n + 1) % (n + 1)] = (f[(count - i + j + n + 1) % (n + 1)] + f[(count + j + n) % (n + 1)] + mod) % mod
	count = (count + i + n + 1) % (n + 1)
print(f[(count - 1 + n) % (n + 1)])


from math import factorial
cat = [1, 1]
p = 10 ** 9 + 7
n = int(input())
ans = 0
fac = [1]
mat = [[0 for i in range(n + 1)] for j in range(n + 1)]
mat[0][0] = 1
for i in range(1, n + 1):
	mat[i][0] = mat[i - 1][i - 1]
	for j in range(i):
		mat[i][j + 1] = (mat[i][j] + mat[i - 1][j]) % p
print(mat[n][n - 1] % p)


sym_trans = [1, 3, 10, 37, 151, 674, 3263, 17007, 94828, 562595, 3535027, 23430840, 163254885, 192059216, 97183539, 384727153, 211932162, 665362848, 415709078, 654959364, 894067657, 839120000, 627072923, 586774033, 440165789, 840647542, 118441546, 505992682, 607243387, 93527420, 378847836, 948139661, 770183251, 365355663, 989344183, 548640276, 317289217, 997901123, 159164073, 843138878, 532150136, 738186543, 903861313, 682196381, 973969945, 132955760, 252975343, 448800686, 260473966, 172364334, 974776377, 897990701, 517201033, 576838003, 99444281, 227668527, 276751933, 925606182, 183971202, 872834804, 675567482, 485151063, 90425806, 573944639, 319383684, 228218188, 853465612, 413776522, 234312776, 696953174, 561359388, 614568659, 310760116, 308156021, 643857743, 130527569, 21545623, 861347933, 160875480, 265384023, 880663120, 642768069, 283279971, 101569565, 642645431, 549146203, 962813505, 943728849, 409246814, 285130248, 71857567, 876844125, 639794356, 718486712, 669604188, 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860988182, 339891870, 990648286, 46638763, 509149276, 541939348, 355904452, 431325298, 357549879, 549943697, 474564051, 162716659, 963090407, 647842561, 772624310, 369470387, 84034776, 592879970, 951519687, 953129351, 88447823, 355811270, 262410819, 122874675, 324874139, 427752611, 966515031, 89544957, 490043087, 514702039, 593866856, 168367087, 644950238, 234858331, 120499562, 727750147, 300432587, 358531606, 837951330, 555717474, 833618568, 749833609, 687809699, 594318929, 382324387, 922817996, 752791004, 138103251, 493853550, 72041429, 300412849, 811511206, 524951630, 444956540, 657742723, 41964071, 115222465, 18151787, 503137364, 742459967, 290436603, 291850913, 707084680, 845615216, 544193854, 838816685, 820365504, 64821808, 958641055, 521245547, 626401031, 826921769, 376015807, 23365585, 127208963, 222201708, 397679385, 704521093, 398003266, 367632312, 560201311, 959095835, 518490388, 782459134, 85411514, 363544205, 898645481, 642423170, 356964076, 218770908, 101398744, 149838090, 847676333, 658844131, 66455878, 967449330, 921735030, 608517334, 501977444, 370764256, 932365136, 842116001, 2414808, 363210962, 468920751, 163861001, 885803673, 639058145, 446132418, 402067874, 794603245, 321367393, 633937175, 752437584, 89831570, 777480614, 1875666, 446948540, 607778609, 895163736, 654319771, 56282607, 953825487, 322366231, 857703755, 40904649, 611285125, 973715621, 685339249, 58869526, 750073307, 286265907, 174980597, 937499408, 885979049, 169542536, 859733362, 993160596, 36179133, 99723628, 300577098, 119273185, 895361237, 321883362, 666141940, 241183990, 611545116, 598039345, 942688158, 648305554, 976484452, 909040395, 639505417, 717880425, 258364578, 177731935, 331433241, 644072879, 835125078, 437946030, 732730195, 263084996, 227710431, 634360769, 949793998, 673705291, 929116367, 780620265, 396385982, 636792029, 981415232, 534698667, 605385771, 446440726, 846286425, 170393385, 865408351, 535702043, 91059914, 396396595, 978353795, 909026952, 193229763, 332588600, 947659377, 659231379, 21658695, 908909731, 930701976, 169444090, 242105422, 211346541, 280700862, 419568817, 565028991, 410179813, 947115493, 162947610, 353283845, 440000302, 555403642, 967699788, 939284378, 26563328, 808244694, 628818450, 464570707, 613515466, 816227814, 798592997, 139960087, 611289279, 84099181, 767531938, 309790621, 725213859, 101876198, 339117512, 560452243, 695943492, 482097635, 696268438, 33261687, 549042129, 227726089, 324472676, 889008129, 433042127, 121540726, 256202561, 466427373, 253057484, 582185803, 12967232, 365976221, 1001525, 843332554, 557829538, 816545092, 17260454, 45247051, 117768503, 97334544, 142496898, 511333733, 432449646, 539910489, 752822850, 319965495, 857056169, 748503998, 709859995, 43212619, 215123872, 763386790, 167320288, 584036346, 431950841, 657125650, 47836908, 931587780, 624620056, 214312432, 650854133, 138350471, 695691723, 818005740, 800549557, 453278187, 704208506, 161679109, 21228857, 24799573, 523890408, 644772103, 703815094, 273840418, 255182503, 451051671, 765402838, 276461433, 817763339, 311617014, 56463857, 173174288, 985489261, 364879760, 260289777, 866138754, 444165753, 814280531, 665038276, 301803455, 959789371, 700429559, 566854653, 152240081, 211426797, 960298, 230714841, 36233184, 269379666, 489824611, 914246569, 385971337, 351582326, 588477371, 891226054, 658857137, 819038771, 551105910, 907928307, 409262069, 270013431, 94369267, 383868816, 550427912, 563194105, 462941349, 484982153, 904457766, 192247167, 381460272, 970265030, 481977727, 31384204, 550046603, 922412185, 814116989, 722990797, 245899471, 637663524, 160142774, 468353543, 438206607, 772758812, 544546780, 544896635, 116902729, 711349637, 991251853, 994312234, 957097261, 412924284, 175915003, 689039777, 818561243, 741321533, 538140579, 281833152, 679071114, 57304296, 833699278, 707474921, 62014389, 120960628, 172686671, 848769176, 927259833, 589949390, 807541116, 181697612, 334825796, 986129478, 59148373, 119891371, 73571388, 74981386, 827459594, 634033443, 904106237, 805784045, 42592013, 13913474, 335897837, 481916336, 598553686, 218288615, 420595899, 842333638, 105897102, 275003777, 283689245, 218844471, 756122698, 252407474, 829890821, 653416861, 368228913, 59776635, 531136583, 47966437, 827773479, 904112376, 898657123, 115622291, 743558, 8828680, 860518048, 777455376, 971624304, 635482863, 511862704, 639631999, 438768263, 901901225, 196727914, 228846332, 683559026, 314126432, 1806760, 214540782, 658500415, 214746490, 608268120, 158011813, 934741554, 940540029, 155900422, 939640865, 651516855, 620660513, 386136335, 745327884, 184835166, 788977403, 243849317, 935550155, 377688755, 32165748, 172100122, 755822381, 999829632, 354024371, 949379942, 411448871, 895625714, 875439951, 926343731, 221188948, 544617857, 290381100, 878249839, 843386003, 120327277, 599441531, 626324309, 222677418, 284201958, 54461302, 576857817, 459925304, 239808803, 254268736, 112223863, 23598162, 687569821, 898753792, 250518185, 143988237, 47361625, 861946417, 779538942, 459568395, 136709445, 556023371, 207252924, 283891715, 777970421, 601703155, 268933499, 562239349, 644182771, 790868861, 383170492, 819558870, 161660156, 71891404, 868527522, 221592101, 545660152, 741329109, 492560139, 333418180, 435874670, 747603335, 289108386, 748335689, 149258814, 947775920, 305799873, 691745662, 859773837, 107234823, 956564497, 619391212, 703591889, 938772236, 739299002, 85626843, 429657017, 880977489, 480485548, 262991906, 6951837, 169090857, 52743047, 668398208, 189604863, 792261239, 245121057, 351830780, 475786287, 298968521, 854649380, 841402304, 705058352, 643408174, 451190291, 106238922, 905773250, 556348580, 833422618, 660705083, 321237184, 837596708, 612787020, 472721787, 312244517, 616099679, 518429803, 244784641, 258499452, 36214998, 60302259, 133990319, 360391216, 503581979, 672913924, 583093439, 550942660, 250979125, 940247510, 652858459, 675878546, 797466841, 370674995, 907745476, 588883865, 987383089, 218448226, 449737002, 955307454, 585163662, 699903733, 105920007, 731369808, 740798623, 732815063, 955069657, 912907186, 197692808, 955042514, 945946070, 849264882, 370954232, 838048808, 299015263, 472283529, 79267470, 314511120, 164725792, 389039359, 173690840, 816568260, 483994943, 820870927, 376726741, 251514316, 51302942, 615552918, 877972951, 860217092, 843254111, 673189570, 27294194, 673071999, 795723031, 758712484, 935436811, 338815770, 883286020, 37036329, 817487284, 363687397, 501326546, 406242075, 412047561, 466907683, 521730345, 887830119, 645973608, 572743108, 627707705, 639840156, 652240338, 640176912, 825286131, 371748226, 535745681, 760583377, 383534127, 366576210, 849032156, 398519118, 755697805, 230126756, 873857996, 596624149, 313413445, 917211514, 268976357, 940260428, 184998150, 343332488, 895030555, 16762969, 293942000, 61558881, 102830492, 276975461, 171586002, 255821416, 397765975, 465541436, 942936259, 901676852, 166714611, 392758236, 685709975, 805075316, 587042608, 808975676, 446219613, 924221182, 717855804, 15861100, 644071222, 797042841, 247404499, 716210868, 160723295, 402081695, 469917635, 28585240, 653821167, 10364285, 208063293, 102556082, 540148245, 666014613, 667552029, 822004563, 166994102, 334528303, 122406032, 697720617, 316059644, 972708995, 726071977, 462361745, 224085543, 572856784, 15957789, 963335493, 927695057, 176379245, 383462849, 865550856, 810104898, 289052741, 20714571, 35324294, 832466091, 743055358, 790404115, 80236396, 795827061, 395369878, 252255602, 184874192, 23530816, 506336207, 658529903, 592654716, 556715519, 822615331, 809520025, 787957548, 139736444, 548229608, 666322128, 72657315, 992131036, 916436220, 768152482, 120580894, 349515729, 528627236, 449845586, 460148110, 371078849, 605223863, 752665040, 1624779, 644242679, 985944307, 858571182, 461684752, 810675957, 993351100, 382520483, 250821716, 977534286, 508168386, 246241034, 609693247, 743320473, 146607662, 998445214, 431759341, 343673635, 640794075, 262660065, 878625869, 897437609, 516489488, 23659927, 759999147, 655412155, 503783506, 667819834, 551718218, 605276486, 57914865, 419059158, 46776049, 804175213, 960309116, 623584323, 753818347, 91180671, 886695514, 802544874, 178110089, 143529344, 505623920, 720477855, 80630633, 63261778, 317677628, 454032332, 240724484, 186782768, 536920726, 899201547, 923187729, 845529381, 295331561, 895582663, 595645523, 82038737, 267747183, 796913715, 122804438, 49897344, 362714249, 862665025, 396997686, 604625458, 37112088, 976519680, 446432499, 748095242, 429003830, 291811017, 677603270, 505978357, 9432426, 387257732, 739252457, 905769562, 480866202, 114477162, 910290154, 964097748, 975047101, 558505018, 397606795, 320449623, 674423851, 567300118, 96600383, 149970539, 81408680, 943126359, 618538215, 902294691, 538833948, 516910200, 175195207, 164991807, 613271994, 282270752, 195159923, 182691277, 893377616, 350962890, 378185065, 165520304, 736097983, 104275527, 107760395, 496223166, 392774298, 339005193, 6415884, 852198248, 869760630, 57337377, 328642311, 313491466, 473943256, 761334845, 163974456, 805443890, 454791777, 655688589, 217832409, 48206873, 250729512, 650171495, 668474483, 645785673, 828763867, 638317152, 787083774, 304210226, 761376754, 998417863, 114437701, 179152011, 856498178, 895998969, 375283462, 756451234, 63585213, 746100628, 352262217, 685049322, 41368875, 185621781, 193116004, 704542713, 613788001, 712282785, 133675350, 36342676, 127889371, 885675751, 150684097, 935926548, 457679454, 908239676, 46086256, 363721489, 148164239, 493754728, 991370249, 574494584, 10015428, 949112564, 670563791, 506987577, 373557253, 422937350, 707248469, 738375556, 170359517, 504522488, 822839646, 971251884, 14295245, 6364559, 829528560, 31902055, 509183036, 190031100, 209870102, 452967833, 456964275, 743009770, 951211379, 933702078, 78834235, 324014730, 139963002, 928413345, 526681054, 322007744, 800710829, 525191165, 830508742, 525334945, 99767615, 190439097, 150011682, 869931254, 195895108, 642716041, 472372818, 845969378, 891314830, 756420050, 48906613, 622902626, 55143732, 131611764, 37731551, 299498677, 640694188, 842798821, 526134613, 706085903, 187560670, 871101937, 458380397, 376491783, 258403652, 497475041, 92698088, 58099292, 798700283, 160237119, 420795881, 715071206, 382414136, 647275638, 598995012, 116058943, 342151526, 611720254, 575486020, 397160465, 573871310, 804763794, 855872322, 662992488, 811636563, 484712697, 785331707, 394682840, 540399090, 628990032, 459093333, 905606245, 300709259, 285160187, 107113222, 57562713, 785917963, 944928720, 215790782, 159729073, 314679186, 484363600, 220810932, 193756864, 176113742, 653956340, 385162015, 613321758, 655032622, 595475307, 963064968, 575934821, 565366446, 339412611, 420269925, 853822178, 621890905, 341799536, 730085337, 669687144, 43038664, 190723760, 520007694, 397191752, 242741101, 295273112, 911553855, 420256891, 134534726, 229041960, 288039667, 912665239, 952930369, 420783608, 759505883, 355080139, 13776242, 725047982, 766014613, 348574940, 279003328, 249479481, 895297500, 190353452, 999650409, 673605968, 308743105, 923509648, 610816753, 868217020, 502437344, 785101081, 683997286, 190190426, 14850091, 628445902, 441409418, 361000413, 482358408, 402504037, 416870733, 249891590, 406197948, 221484197, 989497991, 210224657, 35396951, 560021235, 76667846, 849870082, 830692370, 990522851, 183760025, 844165424, 799741184, 751900362, 312616925, 845225615, 371369299, 912379072, 977692978, 464715793, 272736508, 409713958, 661250225, 355611695, 602025484, 655098052, 791858343, 935074244, 363351667, 660633695, 830681132, 207979827, 766178018, 314428742, 291425335, 249769039, 382582132, 176838020, 531553490, 339637176, 171182908, 867642857, 564528401, 750769745, 600414104, 894717790, 100220991, 558039019, 986383097, 729964739, 105290652, 584230942, 928474519, 731654431, 704603911, 359730029, 568522294, 752842834, 632601136, 5043193, 663508854, 927237848, 469651192, 496701127, 750913782, 564220277, 26335023, 523313806, 906775254, 972081105, 77997445, 356067959, 448714422, 714021267, 864595306, 865545519, 612006278, 769170204, 155531347, 410127755, 951372818, 594730385, 248994712, 771421275, 185158093, 499376011, 662863317, 413001393, 805887756, 473324595, 921297057, 18735813, 386934464, 146972891, 560627347, 903669593, 967094539, 386758849, 737091463, 755426664, 220579995, 435711349, 887741198, 883209694, 478560592, 9832118, 930618930, 163415091, 240446492, 132759678, 134081914, 702601410, 909768461, 660176441, 680440363, 57179043, 900412868, 646477707, 534303308, 997628154, 282709724, 402700228, 997064274, 785650024, 436878044, 977972319, 780587881, 124834909, 341934157]
n = int(input())
print(sym_trans[n - 1])


n = int(input()) + 1
d = 1000000007
g = [[1] * n for i in range(n)]
for i in range(1, n):
	g[i][0] = g[i - 1][i - 1]
	for j in range(1, i + 1):
		g[i][j] = (g[i][j - 1] + g[i - 1][j - 1]) % d
print((g[-1][-1] - g[-1][0]) % d)