A331513 - OEIS

%I #18 May 05 2021 01:53:00

%S 1,4,-6,32,-170,-228,43764,-1498880,43826598,-1249865260,35978752876,

%T -1053020066976,31153402105852,-914722450924436,25562930671296360,

%U -604802562457466880,5868775340572918534,684246820455046681380,-78372285809430441261828

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

%H Seiichi Manyama, <a href="/A331513/b331513.txt">Table of n, a(n) for n = 0..386</a>

%F a(n) = [x^n] (1 + n*x)/(1 + 2*(n-2)*x + (n*x)^2)^(3/2).

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

%t a[n_] := Sum[If[n == n-k == 0, 1, (-n)^(n-k)] * (n+k+1) * Binomial[n, k] * Binomial[n + k, k], {k, 0, n}]; Array[a, 19, 0] (* _Amiram Eldar_, May 05 2021 *)

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

%o (PARI) {a(n) = polcoef((1+n*x)/(1+2*(n-2)*x+(n*x)^2)^(3/2), n)}

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

%Y Cf. A331511, A331512.

%K sign

%O 0,2

%A _Seiichi Manyama_, Jan 19 2020