A390424 Expansion of 1/(2 - g^2)^2, where g = 1+x*g^3 is the g.f. of A001764.

1, 4, 26, 176, 1209, 8360, 57998, 403048, 2803322, 19505448, 135732626, 944461632, 6570687403, 45702062144, 317791588456, 2209127828768, 15352038918434, 106653736269992, 740714729403562, 5142712676327408, 35694581962366591, 247676081665181864, 1718070166963868430

Offset

0,2

Links

Table of n, a(n) for n=0..22.

Formula

G.f.: B(x)^2, where B(x) is the g.f. of A391461.

a(n) = (1/(4*n)) * Sum_{k=1..n} k * ((k+2)*Pell(k+1) + (k+1)*Pell(k+2)) * binomial(3*n,n-k) for n > 0.

Conjecture D-finite with recurrence 10*n*(2*n-5)*a(n) +(-511*n^2+1968*n-1337)*a(n-1) +2*(2461*n^2-12759*n+14814)*a(n-2) +4*(-5447*n^2+35358*n-55483)*a(n-3) +16*(2651*n^2-20323*n+38832)*a(n-4) -2976*(3*n-11)*(3*n-13)*a(n-5)=0. - R. J. Mathar, Jan 26 2026

Prog

(PARI) pell(n) = ([2, 1; 1, 0]^n)[2, 1];
a(n) = if(n==0, 1, sum(k=1, n, k*((k+2)*pell(k+1)+(k+1)*pell(k+2))*binomial(3*n, n-k))/(4*n));

Crossrefs

Cf. A000129, A001764, A391461.

Sequence in context: A210911 A278416 A108082 * A199490 A116429 A392431

Adjacent sequences: A390421 A390422 A390423 * A390425 A390426 A390427

Keyword

nonn

Author

Seiichi Manyama, Dec 10 2025

Status

approved