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L.g.f.: Sum_{n>=1} (x^n/n) * Product_{d|n} (1 + d*x^n).
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%I #13 May 23 2021 06:25:20

%S 1,3,1,7,1,15,1,15,10,13,1,67,1,17,16,63,1,168,1,37,22,25,1,555,1,29,

%T 118,49,1,520,1,543,34,37,1,1048,1,41,40,1125,1,554,1,73,475,49,1,

%U 6651,1,563,52,85,1,1680,1,1457,58,61,1,20632,1,65,787,5087,1

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

%H Antti Karttunen, <a href="/A205479/b205479.txt">Table of n, a(n) for n = 1..2201</a>

%F Forms the logarithmic derivative of A205478.

%e L.g.f.: L(x) = x + 3*x^2/2 + x^3/3 + 7*x^4/4 + x^5/5 + 15*x^6/6 +...

%e By definition:

%e L(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^4)*(1+4*x^4)/4 + x^5*(1+x^5)*(1+5*x^5)/5 + x^6*(1+x^6)*(1+2*x^6)*(1+3*x^6)*(1+6*x^6)/6 +...

%e Exponentiation yields the g.f. of A205478:

%e exp(L(x)) = 1 + x + 2*x^2 + 2*x^3 + 4*x^4 + 4*x^5 + 8*x^6 + 8*x^7 +...

%t max = 70; s = Sum[(x^(n-1)/n)*Product[1+d*x^n, {d, Divisors[n]}], {n, 1, max}] + O[x]^max; CoefficientList[s, x]*Range[max] (* _Jean-François Alcover_, Dec 23 2015 *)

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

%Y Cf. A205478 (exp), A205477, A205481, A205483, A205485, A205487, A205489, A205491.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jan 27 2012