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%I #13 Mar 30 2012 18:37:33
%S 1,3,7,23,51,195,435,1631,4165,14563,34761,141479,327471,1222287,
%T 3267177,11804959,28562075,114349947,272702593,1056583023,2786781123,
%U 9966908779,24678676437,101422669199,243331437901,915276550503,2464145600011,9064045943983,22324762587821
%N L.g.f.: Sum_{n>=1} a(n)*x^n/n = Sum_{n>=1} x^n/n * exp( Sum_{k>=1} a(n*k)*x^(n*k)/k ).
%C L.g.f.: Sum_{n>=1} a(n)*x^n/n = Sum_{n>=1} G_n(x^n)*x^n/n where G_n(x) = exp( Sum_{k>=1} a(n*k)*x^k/k ) are integer series.
%H Paul D. Hanna, <a href="/A203253/b203253.txt">Table of n, a(n) for n = 1..100</a>
%F Equals the logarithmic derivative of A203254.
%e L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 23*x^4/4 + 51*x^5/5 + 195*x^6/6 +...
%e L.g.f.: L(x) = Sum_{n>=1} a(n)*x^n/n = Sum_{n>=1} G_n(x^n)*x^n/n
%e where G_n(x) = exp( Sum_{k>=1} a(n*k)*x^k/k ), which begin:
%e G_1(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 22*x^5 + 62*x^6 + 146*x^7 +...
%e G_2(x) = 1 + 3*x + 16*x^2 + 104*x^3 + 724*x^4 + 5428*x^5 + 44080*x^6 +...;
%e G_3(x) = 1 + 7*x + 122*x^2 + 2128*x^3 + 52330*x^4 + 1109386*x^5 +...;
%e G_4(x) = 1 + 23*x + 1080*x^2 + 67944*x^3 + 4595792*x^4 +...;
%e G_5(x) = 1 + 51*x + 8582*x^2 + 1482524*x^3 + 355949360*x^4 +...;
%e G_6(x) = 1 + 195*x + 89752*x^2 + 53146664*x^3 + 36695632888*x^4 +...;
%e G_7(x) = 1 + 435*x + 705756*x^2 + 1208493276*x^3 +...;
%e G_8(x) = 1 + 1631*x + 7232560*x^2 + 44157620896*x^3 ...; ...
%o (PARI) {a(n)=local(L=vector(n,i,1));for(i=1,n,L=Vec(deriv(sum(m=1,n,x^m/m*exp(sum(k=1,floor(n/m),L[m*k]*x^(m*k)/k)+x*O(x^n))))));L[n]}
%Y Cf. A203254, A209397.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Dec 30 2011
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