A193203 - OEIS

%I #10 Dec 17 2024 16:46:51

%S 1,2,8,136,8256,1405440,602922752,618218766848,1468496379454976,

%T 7913439173996188672,95288304033198904688640,

%U 2534320562857451371729985536,147510069367873486819642346127360,18647939279377935249755798151259226112

%N G.f. satisfies: A(A(x)) = Sum_{n>=1} a(n)*x^n * (1+2^n*x)^n, where g.f. A(x) = Sum_{n>=1} a(n)*x^n.

%e G.f.: A(x) = x + 2*x^2 + 8*x^3 + 136*x^4 + 8256*x^5 + 1405440*x^6 +...

%e where

%e A(A(x)) = x*(1+2*x) + 2*x^2*(1+4*x)^2 + 8*x^3*(1+8*x)^3 + 136*x^4*(1+16*x)^4 + 8256*x^5*(1+32*x)^5 +...+ a(n)*x^n*(1+2^n*x)^n +...

%e Explicitly,

%e A(A(x)) = x + 4*x^2 + 24*x^3 + 360*x^4 + 18496*x^5 + 2939392*x^6 +...

%o (PARI) {a(n)=local(A=[1],F=x,G=x);for(i=1,n,A=concat(A,0);F=x*Ser(A);

%o G=sum(m=1,#A-1,A[m]*x^m*(1+2^m*x+x*O(x^#A))^m);

%o A[#A]=Vec(G)[#A]-Vec(subst(F,x,F))[#A]);if(n<1,0,A[n])}

%Y Cf. A193204.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jul 19 2011