Sum of 4 Consecutive Binomial Coefficients forming Square

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Theorem

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 coefficent
$n$ is an integer
$m$ is a non-negative integer.


Then $n$ has one of the following values:

$-1, 0, 2, 7, 15, 74, 767$

This sequence is A047694 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


The corresponding values of $m$ are:

$0, 1, 2, 8, 24, 260, 8672$

This sequence is A047695 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

\(\displaystyle \) \(\) \(\displaystyle \dbinom {-1} 0 + \dbinom {-1} 1 + \dbinom {-1} 2 + \dbinom {-1} 3\)
\(\displaystyle \) \(=\) \(\displaystyle \left({-1}\right)^0 \dbinom 0 0 + \left({-1}\right)^1 \dbinom 1 1 + \left({-1}\right)^2 \dbinom 2 2 + \left({-1}\right)^3 \dbinom 3 3\) Negated Upper Index of Binomial Coefficient: Corollary 1
\(\displaystyle \) \(=\) \(\displaystyle 1 - 1 + 1 - 1\) Binomial Coefficient with Self
\(\displaystyle \) \(=\) \(\displaystyle 0\)
\(\displaystyle \) \(=\) \(\displaystyle 0^2\)


\(\displaystyle \) \(\) \(\displaystyle \dbinom 0 0 + \dbinom 0 1 + \dbinom 0 2 + \dbinom 0 3\)
\(\displaystyle \) \(=\) \(\displaystyle 1 + 0 + 0 + 0\) Binomial Coefficient with Zero
\(\displaystyle \) \(=\) \(\displaystyle 1\)
\(\displaystyle \) \(=\) \(\displaystyle 1^2\)


\(\displaystyle \) \(\) \(\displaystyle \dbinom 2 0 + \dbinom 2 1 + \dbinom 2 2 + \dbinom 2 3\)
\(\displaystyle \) \(=\) \(\displaystyle 1 + \dbinom 2 1 + \dbinom 2 2 + \dbinom 2 3\) Binomial Coefficient with Zero
\(\displaystyle \) \(=\) \(\displaystyle 1 + 2 + \dbinom 2 2 + \dbinom 2 3\) Binomial Coefficient with One
\(\displaystyle \) \(=\) \(\displaystyle 1 + 2 + 1 + \dbinom 2 3\) Binomial Coefficient with Self
\(\displaystyle \) \(=\) \(\displaystyle 1 + 2 + 1 + 0\) Definition of Binomial Coefficient
\(\displaystyle \) \(=\) \(\displaystyle 4\)
\(\displaystyle \) \(=\) \(\displaystyle 2^2\)


\(\displaystyle \) \(\) \(\displaystyle \dbinom 7 0 + \dbinom 7 1 + \dbinom 7 2 + \dbinom 7 3\)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {7!} {7! \, 0!} + \dfrac {7!} {6! \, 1!} + \dfrac {7!} {5! \, 2!} + \dfrac {7!} {4! \, 3!}\) Definition of Binomial Coefficient
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {7!} {7! \times 1} + \dfrac 7 1 + \dfrac {7 \times 6} {2 \times 1} + \dfrac {7 \times 6 \times 5} {3 \times 2 \times 1}\) Definition of Factorial
\(\displaystyle \) \(=\) \(\displaystyle 1 + 7 + \dfrac {42} 2 + \dfrac {210} 6\)
\(\displaystyle \) \(=\) \(\displaystyle 1 + 7 + 21 + 35\)
\(\displaystyle \) \(=\) \(\displaystyle 64\)
\(\displaystyle \) \(=\) \(\displaystyle 8^2\)


\(\displaystyle \) \(\) \(\displaystyle \dbinom {15} 0 + \dbinom {15} 1 + \dbinom {15} 2 + \dbinom {15} 3\)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {15!} {15! \, 0!} + \dfrac {15!} {14! \, 1!} + \dfrac {15!} {13! \, 2!} + \dfrac {15!} {12! \, 3!}\) Definition of Binomial Coefficient
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {15!} {15! \times 1} + \dfrac {15} 1 + \dfrac {15 \times 14} {2 \times 1} + \dfrac {15 \times 14 \times 13} {3 \times 2 \times 1}\) Definition of Factorial
\(\displaystyle \) \(=\) \(\displaystyle 1 + 15 + \dfrac {210} 2 + \dfrac {2730} 6\)
\(\displaystyle \) \(=\) \(\displaystyle 1 + 15 + 105 + 455\)
\(\displaystyle \) \(=\) \(\displaystyle 576\)
\(\displaystyle \) \(=\) \(\displaystyle 24^2\)


\(\displaystyle \) \(\) \(\displaystyle \dbinom {74} 0 + \dbinom {74} 1 + \dbinom {74} 2 + \dbinom {74} 3\)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {74!} {74! \, 0!} + \dfrac {74!} {73! \, 1!} + \dfrac {74!} {72! \, 2!} + \dfrac {74!} {71! \, 3!}\) Definition of Binomial Coefficient
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {74!} {74! \times 1} + \dfrac {74} 1 + \dfrac {74 \times 73} {2 \times 1} + \dfrac {74 \times 73 \times 72} {3 \times 2 \times 1}\) Definition of Factorial
\(\displaystyle \) \(=\) \(\displaystyle 1 + 74 + \dfrac {5402} 2 + \dfrac {388 \, 944} 6\)
\(\displaystyle \) \(=\) \(\displaystyle 1 + 74 + 2701 + 64 \, 824\)
\(\displaystyle \) \(=\) \(\displaystyle 67 \, 600\)
\(\displaystyle \) \(=\) \(\displaystyle 260^2\)


\(\displaystyle \) \(\) \(\displaystyle \dbinom {767} 0 + \dbinom {767} 1 + \dbinom {767} 2 + \dbinom {767} 3\)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {767!} {767! \, 0!} + \dfrac {767!} {766! \, 1!} + \dfrac {767!} {765! \, 2!} + \dfrac {767!} {764! \, 3!}\) Definition of Binomial Coefficient
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {767!} {767! \times 1} + \dfrac {767} 1 + \dfrac {767 \times 766} {2 \times 1} + \dfrac {767 \times 766 \times 765} {3 \times 2 \times 1}\) Definition of Factorial
\(\displaystyle \) \(=\) \(\displaystyle 1 + 767 + \dfrac {587 \, 522} 2 + \dfrac {449 \, 454 \, 330} 6\)
\(\displaystyle \) \(=\) \(\displaystyle 1 + 767 + 293 \, 761 + 74 \, 909 \, 055\)
\(\displaystyle \) \(=\) \(\displaystyle 75 \, 203 \, 584\)
\(\displaystyle \) \(=\) \(\displaystyle 8672^2\)



Sources