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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.

%I A213336 #15 Oct 03 2023 08:49:44
%S A213336 1,1,8,64,568,5440,54888,574848,6190872,68132224,762874568,8663106496,
%T A213336 99536424952,1155012037824,13516570396968,159340702404352,
%U A213336 1890451582396632,22555522916988672,270466907608087944,3257754635421506368,39397587357527547320
%N 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.
%H A213336 Vincenzo Librandi, <a href="/A213336/b213336.txt">Table of n, a(n) for n = 0..100</a>
%F A213336 G.f. satisfies: A(x) = F(x*A(x)^4) where F(x) = 1 + x/F(-x)^4 is the g.f. of A213335.
%F A213336 G.f. A(x) satisfies: A(1 - G(-x)) = G(x) = 1 + x*G(x)^4 is the g.f. of A002293.
%F A213336 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
%e A213336 G.f.: A(x) = 1 + x + 8*x^2 + 64*x^3 + 568*x^4 + 5440*x^5 + 54888*x^6 +...
%e A213336 G.f.: A(x) = G(x/(1-x)^4) where G(x) = 1 + x*G(x)^4 is g.f. of A002293:
%e A213336 G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +...
%o A213336 (PARI) /* G.f. A(x) = G(x/(1-x)^4) where G(x) = 1 + x*G(x)^4: */
%o A213336 {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)}
%o A213336 for(n=0, 25, print1(a(n), ", "))
%o A213336 (PARI) /* G.f. A(x) = F(x*A(x)^4) where F(x) = 1 + x/F(-x)^4: */
%o A213336 {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)}
%o A213336 for(n=0, 25, print1(a(n), ", "))
%Y A213336 Cf. A213335, A002293; variants: A006319, A213282.
%Y A213336 Partial sums give A349310. - _Seiichi Manyama_, Oct 03 2023
%K A213336 nonn
%O A213336 0,3
%A A213336 _Paul D. Hanna_, Jun 09 2012