A391894 a(n) = Sum_{k=0..n} (k+1) * 2^k * binomial(2*k+1,2*n-2*k+1).

1, 12, 64, 344, 1852, 9504, 47616, 235072, 1145104, 5518400, 26364928, 125049984, 589461440, 2763899904, 12899926016, 59964819456, 277750014208, 1282419653632, 5904300064768, 27113852057600, 124223466028032, 567932306153472, 2591487798673408, 11804042907271168, 53678764743135232

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (8,-16,16,-72,32,-64,64,-16).

Formula

G.f.: ((1-2*x-2*x^2)^2 + 8*x - 16*x^2) / ((1-2*x-2*x^2)^2 - 16*x^3)^2.

a(n) = 8*a(n-1) - 16*a(n-2) + 16*a(n-3) - 72*a(n-4) + 32*a(n-5) - 64*a(n-6) + 64*a(n-7) - 16*a(n-8).

Mathematica

CoefficientList[Series[((1-2*x-2*x^2)^2+8*x-16*x^2)/((1-2*x-2*x^2)^2-16*x^3)^2, {x, 0, 50}], x] (* Vincenzo Librandi, Jan 01 2026 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(((1-2*x-2*x^2)^2+8*x-16*x^2)/((1-2*x-2*x^2)^2-16*x^3)^2)
(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-2*x-2*x^2)^2 + 8*x - 16*x^2) / ((1-2*x-2*x^2)^2 - 16*x^3)^2); // Vincenzo Librandi, Jan 01 2026

Crossrefs

Cf. A390700, A391832, A391895.

Sequence in context: A307061 A380353 A074359 * A104062 A371558 A232383

Adjacent sequences: A391891 A391892 A391893 * A391895 A391896 A391897

Keyword

nonn

Author

Seiichi Manyama, Dec 22 2025

Status

approved