A205484 G.f.: exp( Sum_{n>=1} x^n/n * Product_{d|n} (1 + d*x^d)^n ).

1, 1, 2, 3, 7, 14, 30, 65, 132, 280, 632, 1439, 3299, 7569, 17450, 40313, 92889, 212801, 483590, 1092649, 2467078, 5581232, 12690828, 29123728, 67648617, 159370347, 381080620, 923803158, 2264970530, 5599185887, 13909201590, 34612152762, 86049014990

Offset

0,3

Comments

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

Links

Table of n, a(n) for n=0..32.

Formula

Logarithmic derivative yields A205485.

Example

G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 7*x^4 + 14*x^5 + 30*x^6 + 65*x^7 + ...
By definition:
log(A(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 + ...
Explicitly,
log(A(x)) = x + 3*x^2/2 + 4*x^3/3 + 15*x^4/4 + 31*x^5/5 + 72*x^6/6 + 176*x^7/7 + 327*x^8/8 + 751*x^9/9 + ... + A205485(n)*x^n/n + ...

Mathematica

max = 40; s = Exp[Sum[(x^n/n)*Product[(1 + d*x^d)^n, {d, Divisors[n]}], {n, 1, max}]] + O[x]^max; CoefficientList[s , x] (* Jean-François Alcover, Dec 23 2015 *)

Prog

(PARI) {a(n)=polcoeff(exp(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. A205485 (log), A205476, A205478, A205480, A205482, A205486, A205488, A205490.

Sequence in context: A019595 A112884 A103421 * A151530 A180752 A000642

Adjacent sequences: A205481 A205482 A205483 * A205485 A205486 A205487

Keyword

nonn

Author

Paul D. Hanna, Jan 27 2012

Status

approved