A209397 L.g.f.: Sum_{n>=1} a(n)*x^n/n = Sum_{n>=1} x^n/n * exp( Sum_{k>=1} a(k)*x^(n*k)/k ).

1, 3, 7, 19, 46, 129, 337, 939, 2581, 7238, 20263, 57337, 162319, 461961, 1317217, 3767035, 10792400, 30983565, 89084845, 256531814, 739658815, 2135234247, 6170505666, 17849457873, 51679366171, 149750711581, 434260829464, 1260198317509, 3659410074933

Offset: 1

Author

Paul D. Hanna, Mar 07 2012

Keywords

• nonn

Examples

			L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 19*x^4/4 + 46*x^5/5 + 129*x^6/6 +...
Let G(x) be the g.f. of A000081, then
exp(L(x)) = G(x)/x where G(x) = x*exp( Sum_{n>=1} G(x^n)/n ) begins:
G(x) = x + x^2 + 2*x^3 + 4*x^4 + 9*x^5 + 20*x^6 + 48*x^7 + 115*x^8 + 286*x^9 + 719*x^10 + 1842*x^11 + 4766*x^12 + 12486*x^13 + 32973*x^14 +...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..2136 (first 500 terms from Paul D. Hanna)

Crossrefs

Cf. A000081, A203253.

Programs

PARI

{a(n)=local(L=vector(n, i, 1)); for(i=1, n, L=Vec(deriv(sum(m=1, n, x^m/m*exp(sum(k=1, n\m, L[k]*x^(m*k)/k)+x*O(x^n)))))); L[n]}
for(n=1,30,print1(a(n),","))

Formula

a(n) = Sum_{d|n} d*A000081(d).

L.g.f.: Sum_{n>=1} -A000081(n) * log(1-x^n).

L.g.f.: log( G(x)/x ) = Sum_{n>=1} G(x^n)/n where G(x) is the g.f. of A000081, which is the number of rooted trees with n nodes.

a(n) ~ c * d^n / sqrt(n), where d = A051491 = 2.9557652856519949747148..., c = A187770 = 0.4399240125710253040409... . - Vaclav Kotesovec, Oct 30 2014