A209881 G.f. satisfies: A(x) = 1 + x*[d/dx 1/(1 - x*A(x))].

1, 1, 4, 21, 136, 1030, 8856, 84861, 894928, 10291986, 128165720, 1718395602, 24686953968, 378444958060, 6167922926704, 106525443913245, 1943838547593888, 37375737467294362, 755393226726677976, 16011417246585359046, 355187993770520180400, 8230524179585799932820

Offset: 0

Author

Paul D. Hanna, Mar 14 2012

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 21*x^3 + 136*x^4 + 1030*x^5 + 8856*x^6 +...
The g.f. of A075834, G(x) = 1/(1 - x*A(x)), begins:
G(x) = 1 + x + 2*x^2 + 7*x^3 + 34*x^4 + 206*x^5 + 1476*x^6 +...
The logarithm of the g.f. of A075834 begins:
log(G(x)) = x + x^2/2 + 4*x^3/3 + 21*x^4/4 + 136*x^5/5 + 1030*x^6/6 +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A075834.

Programs

PARI

{a(n)=local(A=1+x); for(i=1, n, A=1+x*deriv(1/(1-x*A+x*O(x^n)))); polcoeff(A, n)}
for(n=0,25,print1(a(n),", "))

Formula

a(n) = n*A075834(n+1) for n>=1. [corrected by Vaclav Kotesovec, Aug 24 2017]

Given g.f. A(x), the g.f. of A075834 = 1 + x/(1 - x*A(x)).

Forms the logarithmic derivative of A075834.

O.g.f. A(x) satisfies: [x^n] ( 1 + x/(1 - x*A(x)) )^(n+1) = (n+1)! for n>=0.

O.g.f. A(x) satisfies: [x^n] exp( n * Integral A(x) dx ) * (n + 1 - A(x)) = 0 for n > 0. - Paul D. Hanna, Jun 04 2018

a(n) ~ exp(-1) * n^2 * n!. - Vaclav Kotesovec, Aug 24 2017