A307091 - OEIS

%I #22 Mar 17 2023 11:29:42

%S 1,1,0,-8,-34,-74,0,736,3334,7606,0,-80464,-372436,-864772,0,9400192,

%T 43976774,103061158,0,-1137528688,-5355697084,-12623082284,0,

%U 140697113792,665238165916,1574005263676,0,-17663830073504,-83769667651816,-198760191043784,0

%N a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(n,2*k)^2.

%H Seiichi Manyama, <a href="/A307091/b307091.txt">Table of n, a(n) for n = 0..1879</a>

%F a(4*n+2) = 0 for n >= 0.

%F From _Peter Bala_, Mar 17 2023: (Start)

%F n*(n-1)*(6*n^2-24*n+23)a(n) = 4*(n-1)*(2*n-3)*(3*n^2-9*n+4)*a(n-1) - 4*(3*n^2-9*n+4)*(2*n-3)^2*a(n-2) - 8*(n-2)*(2*n-3)*(3*n^2-9*n+4)*a(n-3) - 4*(n-2)*(n-3)*(6*n^2-12*n+5)*a(n-4) with a(0) = 1, a(1) = 1, a(2) = 0 and a(3) = -8.

%F a(n) = hypergeom([(1-n)/2, (1-n)/2, -n/2, -n/2], [1/2, 1/2, 1], -1).

%F Conjecture: the supercongruence a(n*p^r) == a(n*p^(r-1)) (mod p^(2*r)) holds for positive integers n and r and all primes p >= 3. (End)

%t Table[Sum[(-1)^k*Binomial[n, 2*k]^2, {k, 0, Floor[n/2]}], {n, 0, 30}] (* _Vaclav Kotesovec_, Mar 24 2019 *)

%t Table[HypergeometricPFQ[{1/2 - n/2, 1/2 - n/2, -n/2, -n/2}, {1/2, 1/2, 1}, -1], {n, 0, 30}] (* _Vaclav Kotesovec_, Mar 24 2019 *)

%o (PARI) {a(n) = sum(k=0, n\2, (-1)^k*binomial(n, 2*k)^2)}

%Y Central coefficients of number triangle A307090.

%Y Cf. A000984, A119358, A119363.

%K sign,easy

%O 0,4

%A _Seiichi Manyama_, Mar 24 2019