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A361716 a(n) = Sum_{k = 0..n-1} (-1)^k*binomial(n,k)^2*binomial(n-1,k).

%I A361716 #39 Oct 06 2023 09:28:26
%S A361716 0,1,-3,-8,45,126,-840,-2400,17325,50050,-378378,-1100736,8576568,
%T A361716 25069968,-199536480,-585307008,4732755885,13919870250,-113936715750,
%U A361716 -335813478000,2775498395670,8194328596740,-68263497731520,-201822515032320
%N A361716 a(n) = Sum_{k = 0..n-1} (-1)^k*binomial(n,k)^2*binomial(n-1,k).
%C A361716 Conjecture: the supercongruence a(n*p^k) == a(n*p^(k-1)) (mod p^(3*k)) holds for all primes p >= 5 and positive integers n and k.
%H A361716 Paolo Xausa, <a href="/A361716/b361716.txt">Table of n, a(n) for n = 0..1000</a>
%H A361716 Wikipedia, <a href="https://en.wikipedia.org/wiki/Dixon%27s_identity">Dixon's identity</a>.
%F A361716 a(n) = Sum_{k = 0..n} (-1)^(n+k) * (k/n) * binomial(n,k)^3.
%F A361716 a(2*n) = (-1)^n * (1/2) * (3*n)!/n!^3 for n >= 1; a(2*n+1) = (-1)^n * (3*n+1)/(2*n+1) * (3n)!/n!^3.
%F A361716 a(2*n) = (1/2)*A245086(2*n) = (1/2)*(-1)^n*A006480(n) for n >= 1.
%F A361716 a(2*n+1) = A361710(2*n+1) = A361711(2*n+1).
%F A361716 a(n) = hypergeom([1 - n, - n, - n], [1, 1], 1) for n >= 1.
%F A361716 P-recursive: n^2*(n-1)*(6*n^2-20*n+17)*a(n) = -( 6*(3*n^2-6*n+2)*(n-1)*a(n-1) + (3*n-6)*(3*n-5)*(3*n-4)*(6*n^2-8*n+3)*a(n-2) ).
%F A361716 a(n) = Sum_{k = 0..n} (-1)^(n+k)*binomial(3*n,n-k)*binomial(n+k,k)^2 for n >= 1.
%F A361716 a(n) = Sum_{k = 0..n-1} (-1)^k*binomial(n,k)*binomial(n+k,n)*binomial(2*n-k-1,n). - _Peter Bala_, Sep 13 2023
%p A361716 seq(add((-1)^k*binomial(n,k)^2*binomial(n-1,k), k = 0..n-1), n = 0..20);
%t A361716 A361716[n_]:=Sum[(-1)^k*Binomial[n,k]^2Binomial[n-1,k],{k,0,n-1}];Array[A361716,30,0] (* _Paolo Xausa_, Oct 06 2023 *)
%o A361716 (PARI) a(n) = sum(k = 0, n-1, (-1)^k*binomial(n,k)^2*binomial(n-1,k)); \\ _Michel Marcus_, Mar 26 2023
%o A361716 (Python)
%o A361716 from math import comb
%o A361716 def A361716(n): return (sum(comb(n,k)**3*k if k&1 else -comb(n,k)**3*k for k in range(n+1)))//(n if n&1 else -n) if n else 0 # _Chai Wah Wu_, Mar 27 2023
%Y A361716 Cf. A006480, A245086, A361710, A361711.
%K A361716 sign,easy
%O A361716 0,3
%A A361716 _Peter Bala_, Mar 23 2023