A308036 - OEIS

%I #19 May 12 2021 03:07:56

%S 1,1,-2,-8,1,61,91,-377,-1469,1027,16120,18250,-132065,-427517,620062,

%T 5707648,3746683,-55581941,-144227438,351490672,2274331579,278638399,

%U -25348074713,-50842003745,195685236757,957135268261,-459626168864,-12027281377922

%N Coefficient of x^n in 1/(n+1) * (1 + x - 3*x^2)^(n+1).

%C Also coefficient of x^n in the expansion of 2/(1 - x + sqrt(1 - 2*x + 13*x^2)).

%H Seiichi Manyama, <a href="/A308036/b308036.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum_{k=0..floor(n/2)} (-3)^k * binomial(n,k) * binomial(n-k,k)/(k+1) = Sum_{k=0..floor(n/2)} (-3)^k * binomial(n,2*k) * A000108(k).

%F (n+2) * a(n) = (2*n+1) * a(n-1) - 13 * (n-1) * a(n-2).

%F a(n) = Hypergeometric2F1(1/2 - n/2, -n/2, 2, -12). - _Vaclav Kotesovec_, May 12 2021

%t a[n_] := Sum[(-3)^k * Binomial[n, 2*k] * CatalanNumber[k], {k, 0, Floor[n/2]}]; Array[a, 28, 0] // Flatten (* _Amiram Eldar_, May 12 2021 *)

%t Table[Hypergeometric2F1[1/2 - n/2, -n/2, 2, -12], {n, 0, 30}] (* _Vaclav Kotesovec_, May 12 2021 *)

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

%o (PARI) {a(n) = sum(k=0, n\2, (-3)^k*binomial(n, k)*binomial(n-k, k)/(k+1))}

%o (PARI) {a(n) = sum(k=0, n\2, (-3)^k*binomial(n, 2*k)*binomial(2*k, k)/(k+1))}

%Y Column 3 of A308035.

%K sign

%O 0,3

%A _Seiichi Manyama_, May 10 2019