A391461 Expansion of 1/(2 - g^2), where g = 1+x*g^3 is the g.f. of A001764.

1, 2, 11, 66, 412, 2630, 17029, 111344, 733202, 4854062, 32269865, 215245074, 1439609582, 9650033224, 64808040028, 435935055728, 2936358225450, 19802076360062, 133678410354017, 903246297123430, 6108010982206984, 41333625143684982, 279888021880711789

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..1193

Formula

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

D-finite with recurrence: (-4752*n^2 - 9504*n - 4224)*a(n) + (6104*n^2 + 15208*n + 9408)*a(n + 1) + (-2042*n^2 - 7516*n - 6672)*a(n + 2) + (265*n^2 + 1337*n + 1608)*a(n + 3) + (-12*n^2 - 78*n - 120)*a(n + 4) = 0. - Robert Israel, Feb 12 2026

Maple

f:= gfun:-rectoproc({(-4752*n^2 - 9504*n - 4224)*a(n) + (6104*n^2 + 15208*n + 9408)*a(n + 1) + (-2042*n^2 - 7516*n - 6672)*a(n + 2) + (265*n^2 + 1337*n + 1608)*a(n + 3) + (-12*n^2 - 78*n - 120)*a(n + 4), a(0) = 1, a(1) = 2, a(2) = 11, a(3) = 66, a(4) = 412}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Feb 12 2026

Prog

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

Crossrefs

Cf. A289684, A391464.

Cf. A000129, A006013.

Sequence in context: A039632 A143816 A220783 * A160852 A185627 A216585

Adjacent sequences: A391458 A391459 A391460 * A391462 A391463 A391464

Keyword

nonn

Author

Seiichi Manyama, Dec 10 2025

Status

approved