A198304 G.f.: exp( Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - n*x^d/d) ).

1, 1, 2, 4, 9, 21, 54, 148, 442, 1433, 5061, 19394, 80308, 357241, 1697870, 8577240, 45845235, 258198133, 1526631800, 9445795717, 60988643813, 409933740177, 2862338202947, 20723903238290, 155329193200741, 1203428108558453, 9624564394649845, 79357873429159078

Offset: 0

Author

Paul D. Hanna, Jan 27 2012

Keywords

• nonn

Comments

Logarithmic derivative yields A198305.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 21*x^5 + 54*x^6 + 148*x^7 +...
such that, by definition:
log(A(x)) = x/(1-x) + (x^2/2)/((1-2*x)*(1-x^2)) + (x^3/3)/((1-3*x)*(1-x^3)) + (x^4/4)/((1-4*x)*(1-2*x^2)*(1-x^4)) + (x^5/5)/((1-5*x)*(1-x^5)) + (x^6/6)/((1-6*x)*(1-3*x^2)*(1-2*x^3)*(1-x^6)) +...+ (x^n/n)/Product_{d|n} (1-n*x^d/d) +...
Explicitly, the logarithm begins:
log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 19*x^4/4 + 51*x^5/5 + 159*x^6/6 + 519*x^7/7 + 1867*x^8/8 + 7234*x^9/9 +...+ A198305(n)*x^n/n +...
		

Crossrefs

Cf. A198305 (log), A198296.

Programs

PARI

{a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*exp(sumdiv(m, d, -log(1-m*x^d/d+x*O(x^n)))))), n)}