A088715 G.f. satisfies: A(x*g(x)) = g(x) where g(x) is the g.f. of A088716.

1, 1, 2, 7, 36, 240, 1926, 17815, 184916, 2116498, 26391700, 355405934, 5134778584, 79178537346, 1297633495518, 22522717498167, 412754532495252, 7965288555078018, 161475849044919996, 3431346397643014818

Offset: 0

Author

Paul D. Hanna, Oct 12 2003

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..440

Crossrefs

Cf. A088716, A238223.

Programs

PARI

a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=exp(sum(k=1,n,(1-x*deriv(log(A)))^(-k)*x^k/k)));polcoeff(A,n) \\ Paul D. Hanna, Aug 31 2009

PARI

a(n)=local(A=1+x); for(i=1, n, A=1+x*A^2/(A-x*deriv(A)+x*O(x^n))); polcoeff(A, n)
for(n=0,30,print1(a(n),", ")) \\ Paul D. Hanna, Mar 20 2013

Formula

G.f.: Coefficient of x^n in A(x)^(n+1)/(n+1) = coefficient of x^n in A(x)^(n+2) = A088716(n).

G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/(1 - x*[A'(x)/A(x)])^n/n ). - Paul D. Hanna, Aug 31 2009

G.f. satisfies: A(x) = 1 + x*A(x)^2/(A(x) - x*A'(x)). - Paul D. Hanna, Mar 20 2013

a(n) ~ c * n! * n^2, where c = A238223 / exp(1) = 0.08017961462469262235245081077906956577... - Vaclav Kotesovec, Feb 21 2014