A205486 G.f.: exp( Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - d*x^(n/d))^d ).

1, 1, 2, 5, 16, 60, 259, 1273, 7048, 43241, 289685, 2097912, 16317134, 135574160, 1196898329, 11168544771, 109647222799, 1128440311914, 12139734936953, 136195813530558, 1590028534430967, 19277087785530470, 242235954813757132, 3149491477171141810

Offset

0,3

Comments

Note: exp( Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - x^(n/d))^d ) does not yield an integer series.

Links

Table of n, a(n) for n=0..23.

Formula

Logarithmic derivative yields A205487.

Example

 G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 16*x^4 + 60*x^5 + 259*x^6 + 1273*x^7 +...
By definition:
log(A(x)) = x/(1-x) + (x^2/2)/((1-x^2)*(1-2*x)^2) + (x^3/3)/((1-x^3)*(1-3*x)^3) + (x^4/4)/((1-x^4)*(1-2*x^2)^2*(1-4*x)^4) + (x^5/5)/((1-x^5)*(1-5*x)^5) + (x^6/6)/((1-x^6)*(1-2*x^3)^2*(1-3*x^2)^3*(1-6*x)^6) +...
Explicitly,
log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 43*x^4/4 + 206*x^5/5 + 1104*x^6/6 + 6581*x^7/7 + 43227*x^8/8 +...+ A205487(n)*x^n/n +...

Prog

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

Crossrefs

Cf. A205487 (log), A205476, A205478, A205480, A205482, A205484, A205488, A205490.

Sequence in context: A352617 A332930 A000764 * A210668 A279564 A005036

Adjacent sequences: A205483 A205484 A205485 * A205487 A205488 A205489

Keyword

nonn

Author

Paul D. Hanna, Jan 27 2012

Status

approved