A203267 L.g.f.: Sum_{n>=1} a(n)*x^n/n = Sum_{n>=1} x^n/n * exp( Sum_{k>=1} 3*a(n*k)*x^(n*k)/k ).

1, 7, 46, 371, 2611, 22444, 163010, 1414763, 10666423, 92901977, 700765693, 6267591344, 47400875250, 421269688378, 3261487427911, 28956966303371, 222519855315655, 2011947117233155, 15451470070634425, 138876292766145541, 1085821838608348370, 9706788507990083429

Offset: 1

Author

Paul D. Hanna, Dec 30 2011

Keywords

• nonn

Comments

L.g.f.: Sum_{n>=1} a(n)*x^n/n = Sum_{n>=1} G_n(x^n)^3 * x^n/n where G_n(x) = exp( Sum_{k>=1} a(n*k)*x^k/k ) are integer series.

Examples

			L.g.f.: L(x) = x + 7*x^2/2 + 46*x^3/3 + 371*x^4/4 + 2611*x^5/5 +...
L.g.f.: L(x) = Sum_{n>=1} a(n)*x^n/n = Sum_{n>=1} G_n(x^n)^3*x^n/n
where G_n(x) = exp( Sum_{k>=1} a(n*k)*x^k/k ), which begin:
G_1(x) = 1 + x + 4*x^2 + 19*x^3 + 116*x^4 + 683*x^5 + 4818*x^6 +...
G_2(x) = 1 + 7*x + 210*x^2 + 8837*x^3 + 427910*x^4 + 22758491*x^5 +...;
G_3(x) = 1 + 46*x + 12280*x^2 + 4087909*x^3 + 1805475734*x^4 +...;
G_4(x) = 1 + 371*x + 776202*x^2 + 2360146453*x^3 +...;
G_5(x) = 1 + 2611*x + 49859649*x^2 + 1211412677799*x^3 +...;
G_6(x) = 1 + 22444*x + 3385662240*x^2 + 742868246890817*x^3 +...;
G_7(x) = 1 + 163010*x + 223920974239*x^2 + 396998122840515180*x^3 +...; ...
		

Crossrefs

Cf. A203268 (exp), A203253, A203265.

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, floor(n/m), 3*L[m*k]*x^(m*k)/k)+x*O(x^n)))))); L[n]}

Formula

Equals the logarithmic derivative of A203268.