A213336 G.f. satisfies A(x) = G(x/(1-x)^4) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

1, 1, 8, 64, 568, 5440, 54888, 574848, 6190872, 68132224, 762874568, 8663106496, 99536424952, 1155012037824, 13516570396968, 159340702404352, 1890451582396632, 22555522916988672, 270466907608087944, 3257754635421506368, 39397587357527547320

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Formula

G.f. satisfies: A(x) = F(x*A(x)^4) where F(x) = 1 + x/F(-x)^4 is the g.f. of A213335.

G.f. A(x) satisfies: A(1 - G(-x)) = G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

a(n) = Sum_{k=0..n} binomial(n+3*k-1,n-k) * binomial(4*k,k)/(3*k+1). - Seiichi Manyama, Oct 03 2023

Example

G.f.: A(x) = 1 + x + 8*x^2 + 64*x^3 + 568*x^4 + 5440*x^5 + 54888*x^6 +...
G.f.: A(x) = G(x/(1-x)^4) where G(x) = 1 + x*G(x)^4 is g.f. of A002293:
G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +...

Prog

(PARI) /* G.f. A(x) = G(x/(1-x)^4) where G(x) = 1 + x*G(x)^4: */
{a(n)=local(A, G=1+x); for(i=1, n, G=1+x*G^4+x*O(x^n)); A=subst(G, x, x/(1-x+x*O(x^n))^4); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* G.f. A(x) = F(x*A(x)^4) where F(x) = 1 + x/F(-x)^4: */
{a(n)=local(F=1+x+x*O(x^n), A=1); for(i=1, n+1, F=1+x/subst(F^4, x, -x+x*O(x^n))); A=(serreverse(x/F^4)/x)^(1/4); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A213335, A002293; variants: A006319, A213282.

Partial sums give A349310. - Seiichi Manyama, Oct 03 2023

Sequence in context: A344054 A344252 A199567 * A366497 A355214 A047900

Adjacent sequences: A213333 A213334 A213335 * A213337 A213338 A213339

Keyword

nonn

Author

Paul D. Hanna, Jun 09 2012

Status

approved