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G.f.: exp( Sum_{n>=1} (x^n/n)/sqrt(1 - 2*(2*x)^n) ).
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%I #9 Mar 30 2012 18:37:25

%S 1,1,3,9,33,115,445,1653,6445,24783,97181,379105,1495607,5884239,

%T 23289639,92143819,365700023,1451737985,5774284819,22976698471,

%U 91541016133,364883522809,1455637611901,5809643314425,23201023852083

%N G.f.: exp( Sum_{n>=1} (x^n/n)/sqrt(1 - 2*(2*x)^n) ).

%F Logarithmic derivative yields A184513.

%e G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 33*x^4 + 115*x^5 + 445*x^6 +...

%e The log of the g.f. equals the series:

%e log(A(x)) = x/sqrt(1-4*x) + (x^2/2)/sqrt(1-8*x^2) + (x^3/3)/sqrt(1-16*x^3) + (x^4/4)/sqrt(1-32*x^4) + (x^5/5)/sqrt(1-64*x^5) +...

%e and may be expressed in terms of the central binomial coefficients (A000984).

%e Explicitly, the logarithm begins:

%e log(A(x)) = x + 5*x^2/2 + 19*x^3/3 + 89*x^4/4 + 351*x^5/5 + 1601*x^6/6 +...

%o (PARI) {a(n)=polcoeff(exp(sum(m=1,n+1,(x^m/m)/sqrt(1-2*(2*x)^m+x*O(x^n)))),n)}

%Y Cf. A184513 (log).

%K nonn

%O 0,3

%A _Paul D. Hanna_, Mar 18 2011