A187827 - OEIS

%I #17 Nov 08 2014 07:25:40

%S 1,1,5,36,325,3468,42519,590268,9201740,160150252,3095440553,

%T 66068011710,1547572760559,39529002357409,1094096683131616,

%U 32622859912512090,1042350065213470532,35521574976088978133,1285782300453328211074,49256935742079848796102

%N G.f. satisfies: A(x) = Sum_{n>=0} (1 - (1 - x*A(x))^n)^n.

%H Vaclav Kotesovec, <a href="/A187827/b187827.txt">Table of n, a(n) for n = 0..140</a>

%F G.f. satisfies: A(x) = Sum_{n>=1} (1-x*A(x))^n * (1 - (1-x*A(x))^n)^(n-1).

%F a(n) ~ c * n^n / (exp(n) * (log(2))^(2*n)), where c = 3.7860088... . - _Vaclav Kotesovec_, Nov 08 2014

%e G.f.: A(x) = 1 + x + 5*x^2 + 36*x^3 + 325*x^4 + 3468*x^5 + 42519*x^6 +...

%e where the g.f. satisfies the identities:

%e (1) A(x) = 1 + x*A(x) + (1 - (1-x*A(x))^2)^2 + (1 - (1-x*A(x))^3)^3 + (1 - (1-x*A(x))^4)^4 + (1 - (1-x*A(x))^5)^5 +...

%e (2) A(x) = (1-x*A(x)) + (1-x*A(x))^2*(1 - (1-x*A(x))^2) + (1-x*A(x))^3*(1 - (1-x*A(x))^3)^2 + (1-x*A(x))^4*(1 - (1-x*A(x))^4)^3 + (1-x)^5*(1 - (1-x*A(x))^5)^4 +...

%o (PARI) {a(n)=local(q, A=1); for(i=1,n,q=1/(1-x*A+x*O(x^n));A=sum(k=0, n+1, q^(-k^2)*(q^k-1)^k)); polcoeff(A, n)}

%o for(n=0,20,print1(a(n),", "))

%o (PARI) {a(n)=local(q, A=1); for(i=1,n,q=1/(1-x*A+x*O(x^n));A=sum(k=1, n+1, q^(-k^2)*(q^k-1)^(k-1))); polcoeff(A, n)}

%o for(n=0,20,print1(a(n),", "))

%Y Cf. A220353, A187826.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 27 2012