cp's OEIS Frontend

A212029 G.f. A(x) satisfies A(x) = 1 + x*A(x*A(x)^3)^3.

%I A212029 #14 Mar 01 2025 08:34:58
%S A212029 1,1,3,21,190,2112,26922,382110,5920788,98862273,1762572957,
%T A212029 33325846461,664774457583,13932829786025,305788481726799,
%U A212029 7008171327166869,167321925537782445,4153009604547937170,106963758805117459392,2854029374011293902121,78773444214057182702790
%N A212029 G.f. A(x) satisfies A(x) = 1 + x*A(x*A(x)^3)^3.
%F A212029 From _Seiichi Manyama_, Mar 01 2025: (Start)
%F A212029 Let a(n,k) = [x^n] A(x)^k.
%F A212029 a(n,0) = 0^n; a(n,k) = k * Sum_{j=0..n} binomial(3*n-3*j+k,j)/(3*n-3*j+k) * a(n-j,3*j). (End)
%e A212029 G.f.: A(x) = 1 + x + 3*x^2 + 21*x^3 + 190*x^4 + 2112*x^5 + 26922*x^6 +...
%e A212029 Related expansions:
%e A212029 A(x)^3 = 1 + 3*x + 12*x^2 + 82*x^3 + 732*x^4 + 7944*x^5 + 99156*x^6 +..
%e A212029 A(x*A(x)^3) = 1 + x + 6*x^2 + 51*x^3 + 560*x^4 + 7155*x^5 + 102495*x^6 +...
%e A212029 A(x*A(x)^3)^3 = 1 + 3*x + 21*x^2 + 190*x^3 + 2112*x^4 + 26922*x^5 +...
%o A212029 (PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1+x*subst(A^3, x, x*A^3)); polcoeff(A, n)}
%o A212029 for(n=0,30,print1(a(n),", "))
%o A212029 (PARI) a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(3*n-3*j+k, j)/(3*n-3*j+k)*a(n-j, 3*j))); \\ _Seiichi Manyama_, Mar 01 2025
%Y A212029 Cf. A143508, A143501, A212028, A381570.
%K A212029 nonn
%O A212029 0,3
%A A212029 _Paul D. Hanna_, Apr 27 2012