A221096 E.g.f. satisfies: A(x) = Sum_{n>=0} log(1 + x*A(x)^(2*n))^n/n!.

1, 1, 4, 42, 768, 19460, 637200, 25724916, 1233957312, 68591031120, 4338982958400, 307907317681920, 24229505587541760, 2094548798610726432, 197370092438311892736, 20140182770328963216000, 2213078753956025271214080, 260601290312643875434817280

Offset: 0

Author

Paul D. Hanna, Jan 01 2013

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 42*x^3/3! + 768*x^4/4! + 19460*x^5/5! +...
where A(x) satisfies:
A(x) = 1 + log(1 + x*A(x)^2) + log(1 + x*A(x)^4)^2/2! + log(1 + x*A(x)^6)^3/3! +...
The e.g.f. also satisfies:
A(x) = 1 + A(x)^2*x + A(x)^4*(A(x)^4-1)*x^2/2! + A(x)^6*(A(x)^6-1)*(A(x)^6-2)*x^3/3! + A(x)^8*(A(x)^8-1)*(A(x)^8-2)*(A(x)^8-3)*x^4/4! +...+ binomial(A(x)^(2*n), n)*x^n +...
		

Crossrefs

Cf. A189981, A221097, A221098, A221099.

Programs

PARI

{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, log(1+x*(A+x*O(x^n))^(2*m))^m/m!)); n!*polcoeff(A, n)}
for(n=0,20,print1(a(n),", "))

PARI

{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, binomial((A+x*O(x^n))^(2*m), m)*x^m)); n!*polcoeff(A, n)}
for(n=0,20,print1(a(n),", "))

PARI

{Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, sum(k=0, m, Stirling1(m, k)*(A+x*O(x^n))^(2*m*k))*x^m/m!)); n!*polcoeff(A, n)}
for(n=0,20,print1(a(n),", "))

Formula

E.g.f. also satisfies:

(1) A(x) = Sum_{n>=0} binomial(A(x)^(2*n), n) * x^n.

(2) A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} Stirling1(n,k) * A(x)^(2*n*k)/n!.