A390700 a(n) = Sum_{k=0..n} (k+1) * 2^k * binomial(2*k,2*(n-k)).

1, 4, 16, 104, 572, 2912, 14720, 73152, 357648, 1729216, 8284928, 39387008, 186030016, 873761792, 4084183040, 19010058240, 88154661120, 407447196672, 1877645414400, 8629807384576, 39568030415872, 181025166598144, 826547003228160, 3767061485273088, 17139894981955584

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

Prog

(PARI) my(A=2, B=1, 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!((1-2*x-2*x^2)^2 + 16*x^3)/((1-2*x-2*x^2)^2 - 16*x^3)^2); // Vincenzo Librandi, Jan 01 2026

Crossrefs

Cf. A108485, A391875.

Sequence in context: A094637 A332783 A332773 * A203716 A330537 A136793

Adjacent sequences: A390697 A390698 A390699 * A390701 A390702 A390703

Keyword

nonn

Author

Seiichi Manyama, Dec 21 2025

Status

approved