A307946 - OEIS

%I #20 May 12 2021 03:07:30

%S 1,-1,2,0,-96,1875,-32184,554631,-9773056,172718325,-2874200000,

%T 35973317666,218394869760,-46968959184459,2890848443624064,

%U -147665402789062500,7121567693920010240,-337669517265832692843,15985827659730523364352,-759295252512454596032456

%N Coefficient of x^n in 1/(n+1) * (1 - n*x - n*x^2)^(n+1).

%C Also coefficient of x^n in the expansion of 2/(1 + n*x + sqrt(1 + 2*n*x + n*(n+4)*x^2)).

%H Seiichi Manyama, <a href="/A307946/b307946.txt">Table of n, a(n) for n = 0..386</a>

%F a(n) = Sum_{k=0..floor(n/2)} (-n)^(n-k) * binomial(n,k) * binomial(n-k,k)/(k+1) = Sum_{k=0..floor(n/2)} (-n)^(n-k) * binomial(n,2*k) * A000108(k).

%F For n>0, a(n) = (-n)^n * Hypergeometric2F1(1/2 - n/2, -n/2, 2, -4/n). - _Vaclav Kotesovec_, May 12 2021

%t a[0] = 1; a[n_] := Sum[(-n)^(n-k) * Binomial[n, 2*k] * CatalanNumber[k], {k, 0, Floor[n/2]}]; Array[a, 20, 0] // Flatten (* _Amiram Eldar_, May 12 2021 *)

%t Join[{1}, Table[(-n)^n * Hypergeometric2F1[1/2 - n/2, -n/2, 2, -4/n], {n, 1, 20}]] (* _Vaclav Kotesovec_, May 12 2021 *)

%o (PARI) {a(n) = polcoef((1-n*x-n*x^2)^(n+1)/(n+1), n)}

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

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

%Y Cf. A000108, A292716, A307911.

%K sign

%O 0,3

%A _Seiichi Manyama_, May 07 2019