A367548 a(n) = Sum_{k = 0..n} binomial(-n, k) * 2^(n - k).

1, 1, 3, -2, 19, -54, 222, -804, 3075, -11630, 44458, -170268, 654766, -2524508, 9758556, -37802952, 146724579, -570450078, 2221230066, -8660901612, 33811886394, -132148736148, 517012584036, -2024632609272, 7935337877454, -31126450260204, 122183595168612

Offset: 0

Author

Peter Luschny, Nov 29 2023

Keywords

• sign

Crossrefs

Cf. A032443.

Programs

Maple

seq(add(binomial(-n, k)*2^(n - k), k = 0..n), n = 0..26);

Mathematica

Table[Sum[Binomial[-n,k]2^(n-k),{k,0,n}],{n,0,30}] (* Harvey P. Dale, Apr 03 2024 *)

Formula

a(n) = 4^n*3^(-n) - binomial(-n, n+1) * hypergeom([1, 2*n+1], [n + 2], -1/2) / 2.

a(n) = [x^n] (3 + 12*x + sqrt(4*x + 1)*(4*x + 3))/(6 + 16*x - 32*x^2).

D-finite with recurrence 9*n*a(n) +6*(6*n-7)*a(n-1) +16*(-n-4)*a(n-2) +32*(-2*n+5)*a(n-3)=0. - R. J. Mathar, Jan 11 2024

From Seiichi Manyama, Jul 30 2025: (Start)

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

a(n) = Sum_{k=0..n} (-1)^k * 3^(n-k) * binomial(2*n,k). (End)