A205482 - OEIS

%I #13 Oct 08 2018 21:07:46

%S 1,1,2,2,5,5,15,15,34,46,84,96,246,282,512,696,1421,1713,3436,4084,

%T 8227,10821,19128,23258,48474,60943,106780,139313,252748,322577,

%U 600660,760872,1365570,1807979,3064882,3951491,7358684,9476993,15962935,21243381

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

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

%F Logarithmic derivative yields A205483.

%e G.f.: A(x) = 1 + x + 2*x^2 + 2*x^3 + 5*x^4 + 5*x^5 + 15*x^6 + 15*x^7 + ...

%e By definition:

%e 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^4)^2*(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^6)^2*(1+3*x^6)^3*(1+6*x^6)^6/6 + ...

%e Explicitly,

%e log(A(x)) = x + 3*x^2/2 + x^3/3 + 11*x^4/4 + x^5/5 + 45*x^6/6 + x^7/7 + 59*x^8/8 + 109*x^9/9 + 53*x^10/10 + ... + A205483(n)*x^n/n + ...

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

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

%Y Cf. A205483 (log), A205476, A205478, A205480, A205484, A205486, A205488, A205490.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 27 2012