A364742 - OEIS

A364742

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

1, 1, 4, 13, 50, 201, 841, 3627, 15993, 71803, 327082, 1508002, 7023446, 32995626, 156173668, 744029238, 3565030063, 17169013899, 83061503584, 403483653745, 1967217524551, 9623463731721, 47220968518786, 232354408276613, 1146254897566224, 5668118931395946

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OFFSET

0,3

LINKS

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

FORMULA

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

G.f.: (1/x) * Series_Reversion( x / (1 + x * (1 + x)^3) ). - Seiichi Manyama, Oct 03 2025

D-finite with recurrence: -18091*(n + 3)*(n + 2)*(n + 1)*a(n) - 9*(5956*n + 18325)*(n + 3)*(n + 2)*a(n + 1) - 3*(n + 3)*(20013*n^2 + 150438*n + 285370)*a(n + 2) - 3*(10872*n^3 + 145611*n^2 + 648613*n + 960356)*a(n + 3) - 3*(2367*n^3 + 41364*n^2 + 236065*n + 441460)*a(n + 4) + 18*(3*n + 19)*(9*n^2 + 42*n - 56)*a(n + 5) + 54*(3*n + 22)*(n + 7)*(3*n + 17)*a(n + 6) = 0. - Robert Israel, Aug 03 2026

MAPLE

with(gfun):

eq:= y*(1-x*(1+x*y)^3)-1:

de:= algeqtodiffeq(eq, y(x)):

rec:= diffeqtorec(de, y(x), a(n)):

f:= rectoproc(rec, a(n), remember):

map(f, [$0..30]); # Robert Israel, Aug 03 2026

PROG

(PARI) a(n) = sum(k=0, n, binomial(n+1, k)*binomial(3*k, n-k))/(n+1);

CROSSREFS

Cf. A001006, A161634, A364743, A364744.

Sequence in context: A149456 A149457 A149458 * A002746 A149459 A149460