A390732 a(n) = Sum_{k=0..n} (k+1) * 2^(n-k) * binomial(2*k,2*(n-k)).

1, 2, 7, 40, 137, 526, 1979, 7092, 25485, 90266, 316415, 1102336, 3813841, 13123366, 44947139, 153295692, 520920213, 1764401586, 5958744327, 20071157144, 67446281689, 226154549822, 756826028235, 2528141873892, 8431100881821, 28073731414730, 93346571969551, 309972436014384

Offset

0,2

Links

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

Index entries for linear recurrences with constant coefficients, signature (4,2,-4,-33,-8,8,32,-16).

Formula

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

a(n) = 4*a(n-1) + 2*a(n-2) - 4*a(n-3) - 33*a(n-4) - 8*a(n-5) + 8*a(n-6) + 32*a(n-7) - 16*a(n-8).

Mathematica

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

Prog

(PARI) my(A=1, B=2, C=4*A^2*B, N=2, M=30, x='x+O('x^M), X=1-A*x-A*B*x^2, Y=3); Vec(sum(k=0, N\2, C^k*binomial(N, 2*k)*X^(N-2*k)*x^(Y*k))/(X^2-C*x^Y)^N)
(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x-2*x^2)^2 + 8*x^3)/((1-x-2*x^2)^2 - 8*x^3)^2); // Vincenzo Librandi, Jan 01 2026

Crossrefs

Cf. A108480, A391876.

Sequence in context: A052443 A153744 A266422 * A189826 A069732 A346964

Adjacent sequences: A390729 A390730 A390731 * A390733 A390734 A390735

Keyword

nonn

Author

Seiichi Manyama, Dec 21 2025

Status

approved