A212391 - OEIS

%I #9 May 16 2012 14:51:35

%S 1,1,3,14,80,516,3608,26729,206808,1655232,13612512,114466491,

%T 980575020,8533242324,75267759072,671721353474,6056517394512,

%U 55104831724236,505422858053560,4669306663437888,43418090784597696,406109012334694211,3818890067546807794

%N a(n) = A212392(n) / n.

%H Paul D. Hanna, <a href="/A212391/b212391.txt">Table of n, a(n) for n = 1..200</a>

%F Given g.f. A(x), then G(x) = d/dx A(x^3)/3 = Sum_{n>=1} n*a(n)*x^(3*n-1) is the g.f. of A212392 and satisfies: G(x) = (x + G(G(x)))^2.

%F G.f. satisfies: A’(x) = ( 1 + x*A’(x)^2 * A’(x^2*A’(x)^3) )^2 where A'(x) = d/dx A(x).

%e G.f.: A(x) = x + x^2 + 3*x^3 + 14*x^4 + 80*x^5 + 516*x^6 + 3608*x^7 + 26729*x^8 +...

%e Let G(x) = d/dx A(x^3)/3, then G(x) = (x + G(G(x)))^2, where

%e G(x) = x^2 + 2*x^5 + 9*x^8 + 56*x^11 + 400*x^14 + 3096*x^17 + 25256*x^20 +...

%e G(G(x)) = x^4 + 4*x^7 + 24*x^10 + 168*x^13 + 1284*x^16 + 10384*x^19 +...

%o (PARI) {a(n)=local(G=x^2+x^3);for(i=1,n,G=(x+subst(G,x,G+O(x^(3*n))))^2);polcoeff(G,3*n-1)/n}

%o for(n=1,30,print1(a(n),", "))

%Y Cf. A212392.

%K nonn

%O 1,3

%A _Paul D. Hanna_, May 12 2012