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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.
10
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
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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 09 2012
STATUS
approved