A187826 G.f. satisfies: A(x) = Sum_{n>=0} ((1 + x*A(x))^n - 1)^n / (1 + x*A(x))^(n^2).

1, 1, 4, 26, 219, 2227, 26438, 359904, 5555201, 96383191, 1864908541, 39929905561, 938897407239, 24069888638463, 668309231078015, 19977542570492051, 639571311256259372, 21828488143257352752, 791044181963746918758, 30331001954496565907536

Offset: 0

Author

Paul D. Hanna, Dec 27 2012

Keywords

• nonn

Comments

Limit n->infinity A220353(n)/A187826(n) = 1. - Vaclav Kotesovec, Nov 08 2014

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 26*x^3 + 219*x^4 + 2227*x^5 + 26438*x^6 +...
where the g.f. satisfies the identities:
(1) A(x) = 1 + x*A(x)/(1+x*A(x)) + ((1 + x*A(x))^2 - 1)^2/(1+x*A(x))^4 + ((1 + x*A(x))^3 - 1)^3/(1+x*A(x))^9 + ((1 + x*A(x))^4 - 1)^4/(1+x*A(x))^16 +...
(2) A(x) = 1/(1+x*A(x)) + ((1 + x*A(x))^2 - 1)/(1+x*A(x))^4 + ((1 + x*A(x))^3 - 1)^2/(1+x*A(x))^9 + ((1 + x*A(x))^4 - 1)^3/(1+x*A(x))^16 +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..140

Crossrefs

Cf. A220352, A187827, A220353.

Programs

PARI

{a(n)=local(q, A=1); for(i=1,n,q=1+x*A+x*O(x^n);A=sum(k=0, n+1, q^(-k^2)*(q^k-1)^k)); polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))

PARI

{a(n)=local(q, A=1); for(i=1,n,q=1+x*A+x*O(x^n);A=sum(k=1, n+1, q^(-k^2)*(q^k-1)^(k-1))); polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f. satisfies: A(x) = Sum_{n>=1} ((1+x*A(x))^n - 1)^(n-1) / (1+x*A(x))^(n^2).

a(n) ~ c * n^n / (exp(n) * (log(2))^(2*n)), where c = 2.341658334181687683758... . - Vaclav Kotesovec, Nov 08 2014

In closed form, c = 1 / (log(2) * sqrt(1-log(2)) * 2^((1-log(2))/2)). - Vaclav Kotesovec, May 03 2015