Theorem: Sum of 4 Consecutive Binomial Coefficients forming Square

Consider the Diophantine equation:
:$\dbinom n 0 + \dbinom n 1 + \dbinom n 2 + \dbinom n 3 = m^2$
where:
:$\dbinom a b$ denotes a binomial coefficient
:$n$ is an integer
:$m$ is a non-negative integer.
Then $n$ has one of the following values:
:$-1, 0, 2, 7, 15, 74, 767$
{{OEIS|A047694}}
The corresponding values of $m$ are:
:$0, 1, 2, 8, 24, 260, 8672$
{{OEIS|A047695}}

Proof:
{{begin-eqn}}
{{eqn | r = \dbinom {-1} 0 + \dbinom {-1} 1 + \dbinom {-1} 2 + \dbinom {-1} 3
      | o = 
      | c = 
}}
{{eqn | r = \left({-1}\right)^0 \dbinom 0 0 + \left({-1}\right)^1 \dbinom 1 1 + \left({-1}\right)^2 \dbinom 2 2 + \left({-1}\right)^3 \dbinom 3 3
      | c = Negated Upper Index of Binomial Coefficient: Corollary 1
}}
{{eqn | r = 1 - 1 + 1 - 1
      | c = Binomial Coefficient with Self
}}
{{eqn | r = 0
      | c = 
}}
{{eqn | r = 0^2
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | r = \dbinom 0 0 + \dbinom 0 1 + \dbinom 0 2 + \dbinom 0 3
      | o = 
      | c = 
}}
{{eqn | r = 1 + 0 + 0 + 0
      | c = Binomial Coefficient with Zero
}}
{{eqn | r = 1
      | c = 
}}
{{eqn | r = 1^2
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | r = \dbinom 2 0 + \dbinom 2 1 + \dbinom 2 2 + \dbinom 2 3
      | o = 
      | c = 
}}
{{eqn | r = 1 + \dbinom 2 1 + \dbinom 2 2 + \dbinom 2 3
      | c = Binomial Coefficient with Zero
}}
{{eqn | r = 1 + 2 + \dbinom 2 2 + \dbinom 2 3
      | c = Binomial Coefficient with One
}}
{{eqn | r = 1 + 2 + 1 + \dbinom 2 3
      | c = Binomial Coefficient with Self
}}
{{eqn | r = 1 + 2 + 1 + 0
      | c = {{Defof|Binomial Coefficient}}
}}
{{eqn | r = 4
      | c = 
}}
{{eqn | r = 2^2
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | r = \dbinom 7 0 + \dbinom 7 1 + \dbinom 7 2 + \dbinom 7 3
      | o = 
      | c = 
}}
{{eqn | r = \dfrac {7!} {7! \, 0!} + \dfrac {7!} {6! \, 1!} + \dfrac {7!} {5! \, 2!} + \dfrac {7!} {4! \, 3!}
      | c = {{Defof|Binomial Coefficient}}
}}
{{eqn | r = \dfrac {7!} {7! \times 1} + \dfrac 7 1 + \dfrac {7 \times 6} {2 \times 1} + \dfrac {7 \times 6 \times 5} {3 \times 2 \times 1}
      | c = {{Defof|Factorial}}
}}
{{eqn | r = 1 + 7 + \dfrac {42} 2 + \dfrac {210} 6
      | c = 
}}
{{eqn | r = 1 + 7 + 21 + 35
      | c = 
}}
{{eqn | r = 64
      | c = 
}}
{{eqn | r = 8^2
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | r = \dbinom {15} 0 + \dbinom {15} 1 + \dbinom {15} 2 + \dbinom {15} 3
      | o = 
      | c = 
}}
{{eqn | r = \dfrac {15!} {15! \, 0!} + \dfrac {15!} {14! \, 1!} + \dfrac {15!} {13! \, 2!} + \dfrac {15!} {12! \, 3!}
      | c = {{Defof|Binomial Coefficient}}
}}
{{eqn | r = \dfrac {15!} {15! \times 1} + \dfrac {15} 1 + \dfrac {15 \times 14} {2 \times 1} + \dfrac {15 \times 14 \times 13} {3 \times 2 \times 1}
      | c = {{Defof|Factorial}}
}}
{{eqn | r = 1 + 15 + \dfrac {210} 2 + \dfrac {2730} 6
      | c = 
}}
{{eqn | r = 1 + 15 + 105 + 455
      | c = 
}}
{{eqn | r = 576
      | c = 
}}
{{eqn | r = 24^2
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | r = \dbinom {74} 0 + \dbinom {74} 1 + \dbinom {74} 2 + \dbinom {74} 3
      | o = 
      | c = 
}}
{{eqn | r = \dfrac {74!} {74! \, 0!} + \dfrac {74!} {73! \, 1!} + \dfrac {74!} {72! \, 2!} + \dfrac {74!} {71! \, 3!}
      | c = {{Defof|Binomial Coefficient}}
}}
{{eqn | r = \dfrac {74!} {74! \times 1} + \dfrac {74} 1 + \dfrac {74 \times 73} {2 \times 1} + \dfrac {74 \times 73 \times 72} {3 \times 2 \times 1}
      | c = {{Defof|Factorial}}
}}
{{eqn | r = 1 + 74 + \dfrac {5402} 2 + \dfrac {388 \, 944} 6
      | c = 
}}
{{eqn | r = 1 + 74 + 2701 + 64 \, 824
      | c = 
}}
{{eqn | r = 67 \, 600
      | c = 
}}
{{eqn | r = 260^2
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | r = \dbinom {767} 0 + \dbinom {767} 1 + \dbinom {767} 2 + \dbinom {767} 3
      | o = 
      | c = 
}}
{{eqn | r = \dfrac {767!} {767! \, 0!} + \dfrac {767!} {766! \, 1!} + \dfrac {767!} {765! \, 2!} + \dfrac {767!} {764! \, 3!}
      | c = {{Defof|Binomial Coefficient}}
}}
{{eqn | r = \dfrac {767!} {767! \times 1} + \dfrac {767} 1 + \dfrac {767 \times 766} {2 \times 1} + \dfrac {767 \times 766 \times 765} {3 \times 2 \times 1}
      | c = {{Defof|Factorial}}
}}
{{eqn | r = 1 + 767 + \dfrac {587 \, 522} 2 + \dfrac {449 \, 454 \, 330} 6
      | c = 
}}
{{eqn | r = 1 + 767 + 293 \, 761 + 74 \, 909 \, 055
      | c = 
}}
{{eqn | r = 75 \, 203 \, 584
      | c = 
}}
{{eqn | r = 8672^2
      | c = 
}}
{{end-eqn}}
{{ProofWanted|It remains to show there are no more.}}