This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A273630 #18 Oct 04 2022 18:23:42
%S A273630 0,-1,0,162,0,-11250,0,576240,0,-25259850,0,1007242236,0,-37685439792,
%T A273630 0,1346871240000,0,-46504059326010,0,1562983866658500,0,
%U A273630 -51407781284599740,0,1661123953798807680,0,-52886433789393750000,0,1662782404368229351200
%N A273630 a(n) = Sum_{k = 0..n} (-1)^k*k^3*binomial(n,k)^3.
%C A273630 Let d(n) = Sum_{k = 0..n} (-1)^k*binomial(n,k)^3. Clearly, by symmetry of the binomial coefficients we have d(2*n + 1) = 0. Dixon's identity is the result d(2*n) = (-1)^n*(3*n)!/n!^3. A generalization is: for r a nonnegative integer there holds Sum_{k = 0..n} (-1)^k*binomial(k,r)^3*binomial(n,k)^3 = (-1)^r*binomial(n,r)^3*d(n - r). This is the case r = 1. See A273631 (case r = 2) and A245086 (case r = 0).
%H A273630 Peter Bala, <a href="/A273630/a273630.txt">A generalization of Dixon's identity</a>
%H A273630 J. Ward, <a href="http://www.maths.tcd.ie/pub/ims/bull27/bull27_46-54.pdf">100 Years of Dixon's Identity</a>, Irish Mathematical Society Bulletin 27, 46-54, 1991
%H A273630 Wikipedia, <a href="https://en.wikipedia.org/wiki/Dixon%27s_identity">Dixon's identity</a>
%F A273630 a(2*n) = 0; a(2*n + 1) = (-1)^(n+1)*(2*n + 1)^3*(3*n)!/n!^3.
%F A273630 a(2*n + 1) = -(2*n + 1)^3*A245086(2*n) = (-1)^(n+1)* (2*n + 1)^3*A006480(n).
%F A273630 a(n) = Sum_{k = 1..n} (-1)^k*multinomial(n, 1, k - 1, n - k)^3.
%F A273630 Recurrence: a(n) = -3*n^3*(3*n - 5)*(3*n - 7)/((n - 1)^2*(n - 2)^3) * a(n-2).
%p A273630 seq(add((-1)^k*k^3*binomial(n,k)^3, k = 0..n), n = 0..30);
%t A273630 Table[Sum[(-1)^k*k^3 Binomial[n, k]^3, {k, 0, n}], {n, 0, 27}] (* _Michael De Vlieger_, Jul 22 2016 *)
%o A273630 (PARI) a(n) = sum(k=0, n, (-1)^k*k^3*binomial(n, k)^3) \\ _Felix Fröhlich_, Jul 22 2016
%o A273630 (Magma) [&+[(-1)^k*k^3 *Binomial(n, k)^3: k in [0..n]]: n in [0..70]]; // _Vincenzo Librandi_, Jul 23 2016
%o A273630 (Python)
%o A273630 from math import factorial
%o A273630 def A273630(n): return (1 if (m:=n>>1)&1 else -1)*n**3*factorial(3*m)//factorial(m)**3 if n&1 else 0 # _Chai Wah Wu_, Oct 04 2022
%Y A273630 Cf. A006480, A245086, A273631.
%K A273630 sign,easy
%O A273630 0,4
%A A273630 _Peter Bala_, Jul 17 2016