A366327 G.f. satisfies A(x) = (1 + x) * (1 + x/A(x)^3).

1, 2, -5, 33, -260, 2263, -20979, 203124, -2030121, 20786694, -216928144, 2298911699, -24673591005, 267644087524, -2929602893537, 32317666058508, -358931896710948, 4010200327457883, -45040693394259858, 508253687784232108, -5759468659295939684

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..920

Formula

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

D-finite with recurrence: 12*(n - 1)*(3*n - 1)*(3*n + 1)*a(n) + 6*(15*n - 1)*(3*n + 2)*(2*n + 1)*a(n + 1) + (2023*n^3 + 4856*n^2 + 2961*n + 480)*a(n + 2) + (2750*n^3 + 12211*n^2 + 13245*n + 1464)*a(n + 3) + (121*n^3 - 4171*n^2 - 26676*n - 36600)*a(n + 4) - 4*(428*n^3 + 5896*n^2 + 25335*n + 33222)*a(n + 5) - (391*n^3 + 5150*n^2 + 19047*n + 13812)*a(n + 6) + (310*n^3 + 6827*n^2 + 49941*n + 121380)*a(n + 7) + 3*(3*n + 22)*(n + 8)*(3*n + 23)*a(n + 8) = 0. - Robert Israel, Jun 07 2026

Maple

f:= gfun:-rectoproc({12*(n - 1)*(3*n - 1)*(3*n + 1)*a(n) + 6*(15*n - 1)*(3*n + 2)*(2*n + 1)*a(n + 1) + (2023*n^3 + 4856*n^2 + 2961*n + 480)*a(n + 2) + (2750*n^3 + 12211*n^2 + 13245*n + 1464)*a(n + 3) + (121*n^3 - 4171*n^2 - 26676*n - 36600)*a(n + 4) - 4*(428*n^3 + 5896*n^2 + 25335*n + 33222)*a(n + 5) - (391*n^3 + 5150*n^2 + 19047*n + 13812)*a(n + 6) + (310*n^3 + 6827*n^2 + 49941*n + 121380)*a(n + 7) + 3*(3*n + 22)*(n + 8)*(3*n + 23)*a(n + 8), a(0) = 1, a(1) = 2, a(2) = -5, a(3) = 33, a(4) = -260, a(5) = 2263, a(6) = -20979, a(7) = 203124}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Jun 07 2026

Prog

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

Crossrefs

Cf. A073157, A364336, A364337, A364338, A364339, A366325, A366326, A366328.

Cf. A006632, A364407.

Sequence in context: A019037 A019038 A198089 * A197974 A208985 A333738

Adjacent sequences: A366324 A366325 A366326 * A366328 A366329 A366330

Keyword

sign

Author

Seiichi Manyama, Oct 07 2023

Status

approved