A205480 G.f: exp( Sum_{n>=1} x^n/n * Product_{d|n} (1 + d*x^(n/d))^d ).

1, 1, 2, 4, 10, 27, 76, 242, 852, 3016, 11262, 47004, 204761, 894673, 4134909, 20370101, 101904474, 521459464, 2813783214, 15616060213, 87143803196, 502477538546, 3039137586808, 18763942581733, 116737580008529, 742909490860950, 4846956807516551

Offset: 0

Author

Paul D. Hanna, Jan 27 2012

Keywords

• nonn

Comments

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

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 27*x^5 + 76*x^6 + 242*x^7 + ...
By definition:
log(A(x)) = x*(1+x) + x^2*(1+x^2)*(1+2*x)^2/2 + x^3*(1+x^3)*(1+3*x)^3/3 + x^4*(1+x^4)*(1+2*x^2)^2*(1+4*x)^4/4 + x^5*(1+x^5)*(1+5*x)^5/5 + x^6*(1+x^6)*(1+2*x^3)^2*(1+3*x^2)^3*(1+6*x)^6/6 + ...
Explicitly,
log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 23*x^4/4 + 76*x^5/5 + 249*x^6/6 + 974*x^7/7 + 4151*x^8/8 + 16558*x^9/9 + ... + A205481(n)*x^n/n + ...
		

Crossrefs

Cf. A205481 (log), A205476, A205478, A205482, A205484, A205486, A205488, A205490.

Programs

Mathematica

max = 30; s = Exp[Sum[(x^n/n)*Product[(1+d*x^(n/d))^d, {d, Divisors[n]}], {n, 1, max}]] + O[x]^max; CoefficientList[s , x] (* Jean-François Alcover, Dec 23 2015 *)

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)}

Formula

Logarithmic derivative yields A205481.