A205485 L.g.f.: Sum_{n>=1} x^n/n * Product_{d|n} (1 + d*x^d)^n.

1, 3, 4, 15, 31, 72, 176, 327, 751, 2063, 5138, 12708, 30993, 75386, 182644, 433255, 1004854, 2279349, 5115960, 11580835, 26533616, 62024966, 149683357, 373141332, 957942931, 2516465279, 6694846987, 17883365774, 47644695777, 125952933062, 329364348277

Offset

1,2

Links

Table of n, a(n) for n=1..31.

Formula

Forms the logarithmic derivative of A205484.

Example

 L.g.f.: L(x) = x + 3*x^2/2 + 4*x^3/3 + 15*x^4/4 + 31*x^5/5 + 72*x^6/6 +...
By definition:
L(x) = x*(1+x) + x^2*(1+x)^2*(1+2*x^2)^2/2 + x^3*(1+x)^3*(1+3*x^3)^3/3 + x^4*(1+x)^4*(1+2*x^2)^4*(1+4*x^4)^4/4 + x^5*(1+x)^5*(1+5*x^5)^5/5 + x^6*(1+x)^6*(1+2*x^2)^6*(1+3*x^3)^6*(1+6*x^6)^6/6 +...
Exponentiation yields the g.f. of A205484:
exp(L(x)) = 1 + x + 2*x^2 + 3*x^3 + 7*x^4 + 14*x^5 + 30*x^6 + 65*x^7 +...

Prog

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

Crossrefs

Cf. A205484 (exp), A205477, A205479, A205481, A205483, A205487, A205489, A205491.

Sequence in context: A024858 A293047 A135100 * A344261 A055486 A041665

Adjacent sequences: A205482 A205483 A205484 * A205486 A205487 A205488

Keyword

nonn

Author

Paul D. Hanna, Jan 27 2012

Status

approved