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%I #2 Mar 30 2012 18:37:16
%S 1,2,14,268,21462,7872396,12585797612,84949155244024,
%T 2379063526056509734,273414369715003663482380,
%U 128009001272184822673783879332,242979321424122460096958142064785384
%N G.f.: A(x) = exp( Sum_{n>=1} 2^(n^2)/(1 - 2^n*x)^n * x^n/n ).
%C An example of this logarithmic identity at q=2:
%C Sum_{n>=1} [q^(n^2)/(1 - q^n*x)^n]*x^n/n = Sum_{n>=1} [(1 + q^n)^n - 1]*x^n/n.
%F G.f.: A(x) = (1-x)*exp( Sum_{n>=1} (1 + 2^n)^n * x^n/n );
%F Equals the first differences of A155201.
%e G.f.: A(x) = 1 + 2*x + 14*x^2 + 268*x^3 + 21462*x^4 +...
%e log(A(x)) = 2/(1-2*x)*x + 2^4/(1-2^2*x)^2*x^2/2 + 2^9/(1-2^3*x)^3*x^3/3 +...
%e log(A(x)) = (3-1)*x + (5^2-1)*x^2/2 + (9^3-1)*x^3/3 + (17^4-1)*x^4/4 +...
%o (PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, 2^(m^2)/(1-2^m*x)^m*x^m/m)+x*O(x^n)), n)}
%o (PARI) /* As First Differences of A155201: */
%o {a(n)=polcoeff((1-x)*exp(sum(m=1, n+1, (2^m+1)^m*x^m/m)+x*O(x^n)), n)}
%Y Cf. A155201, A155200.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Feb 17 2009
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