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%I #18 Oct 04 2022 18:22:27
%S 0,0,1,0,-1296,0,303750,0,-36879360,0,3157481250,0,-217564322976,0,
%T 12926105848656,0,-689598074880000,0,33901459248661290,0,
%U -1562983866658500000,0,68423756889802253940,0,-2870422192164339671040,0,116191495035298068750000
%N a(n) = Sum_{k = 0..n} (-1)^k*binomial(k,2)^3*binomial(n,k)^3.
%C 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 = 2. See A273630 (case r = 1) and A245086 (case r = 0).
%H Peter Bala, <a href="/A273630/a273630.txt">A generalization of Dixon's identity</a>
%H 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 Wikipedia, <a href="https://en.wikipedia.org/wiki/Dixon%27s_identity">Dixon's identity</a>
%F a(0) = 0 and a(2*n + 2) = (-1)^n*binomial(2*n + 2,2)^3*(3*n)!/n!^3 for n >= 0. a(2*n + 1) = 0.
%F a(2*n + 2) = (-1)^n*(n + 1)^3*(2*n + 1)^3 * A006480(n) for n >= 0.
%F a(n) = Sum_{k = 2..n} (-1)^k*multinomial(n, 2, k - 2, n - k)^3.
%F Recurrence: a(n) = -3*n^3*(n - 1)^3*(3*n - 8)*(3*n - 10)/((n - 2)^5*(n - 3)^3) * a(n-2).
%p seq(add((-1)^k*binomial(k,2)^3*binomial(n,k)^3, k = 0..n), n = 0..30);
%t Table[Sum[(-1)^k*Binomial[k, 2]^3 Binomial[n, k]^3, {k, 0, n}], {n, 0, 27}] (* _Michael De Vlieger_, Jul 22 2016 *)
%o (PARI) a(n) = sum(k=0, n, (-1)^k*binomial(k, 2)^3*binomial(n, k)^3) \\ _Felix Fröhlich_, Jul 22 2016
%o (Magma) [&+[(-1)^k*Binomial(k,2)^3*Binomial(n,k)^3: k in [0..n]]: n in [0..70]]; // _Vincenzo Librandi_, Jul 23 2016
%o (Python)
%o from math import factorial
%o def A273631(n): return 0 if n&1 or n == 0 else (-1 if (m:=n-1>>1)&1 else 1)*((m+1)*(n-1))**3*factorial(3*m)//factorial(m)**3 # _Chai Wah Wu_, Oct 04 2022
%Y Cf. A006480, A245086, A273630.
%K sign,easy
%O 0,5
%A _Peter Bala_, Jul 17 2016
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