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 A203265 L.g.f.: Sum_{n>=1} a(n)*x^n/n = Sum_{n>=1} x^n/n * exp( Sum_{k>=1} 2*a(n*k)*x^(n*k)/k ). 2

%I #8 Mar 30 2012 18:37:33

%S 1,5,22,125,576,3554,16843,103917,521338,3189600,15813205,101516930,

%T 501568809,3154939135,16288999167,101770328205,513944896547,

%U 3322082384450,16707380500562,106553006536680,554390049927421,3479202589748077,17774723219041838

%N L.g.f.: Sum_{n>=1} a(n)*x^n/n = Sum_{n>=1} x^n/n * exp( Sum_{k>=1} 2*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)^2 * x^n/n where G_n(x) = exp( Sum_{k>=1} a(n*k)*x^k/k ) are integer series.

%F Equals the logarithmic derivative of A203266.

%e L.g.f.: L(x) = x + 5*x^2/2 + 22*x^3/3 + 125*x^4/4 + 576*x^5/5 + 3554*x^6/6 +...

%e L.g.f.: L(x) = Sum_{n>=1} a(n)*x^n/n = Sum_{n>=1} G_n(x^n)^2*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 + 3*x^2 + 10*x^3 + 43*x^4 + 172*x^5 + 852*x^6 +...

%e G_2(x) = 1 + 5*x + 75*x^2 + 1518*x^3 + 34663*x^4 + 867760*x^5 +...;

%e G_3(x) = 1 + 22*x + 2019*x^2 + 214648*x^3 + 31221037*x^4 +...;

%e G_4(x) = 1 + 125*x + 59771*x^2 + 40659310*x^3 + 31438395303*x^4 +...;

%e G_5(x) = 1 + 576*x + 1760688*x^2 + 6380121685*x^3 +...;

%e G_6(x) = 1 + 3554*x + 57073923*x^2 + 1295238092004*x^3 +...;

%e G_7(x) = 1 + 16843*x + 1719312892*x^2 + 212162358939394*x^3 +...;

%e G_8(x) = 1 + 103917*x + 56284535547*x^2 + 44125115136389518*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), 2*L[m*k]*x^(m*k)/k)+x*O(x^n)))))); L[n]}

%Y Cf. A203266 (exp), A203253, A203267.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Dec 30 2011

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