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G.f.: exp( Sum_{n>=1} x^n/n * exp( Sum_{k>=1} binomial(2*n*k,n*k)/2 * x^(n*k)/k ) ).
2

%I #5 Mar 30 2012 18:37:34

%S 1,1,2,4,11,27,92,252,906,2787,10191,31594,125998,393021,1535964,

%T 5161328,20221291,66306664,273756969,897440988,3664037417,12621555612,

%U 50496343297,170909672792,725703552284,2427269270146,9982179588261,35179417316991,143999051236064

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

%F G.f.: exp( Sum_{n>=1} C_n(x^n) * x^n/n ) where C_n(x^n) = Product_{k=0..n-1} C( exp(2*Pi*I*k/n)*x ), where C(x) is the Catalan function (A000108).

%e G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 11*x^4 + 27*x^5 + 92*x^6 +...

%e By definition:

%e log(A(x)) = (1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 +...)*x

%e + (1 + 3*x^2 + 22*x^4 + 211*x^6 + 2306*x^8 + 27230*x^10 +...)*x^2/2

%e + (1 + 10*x^3 + 281*x^6 + 10580*x^9 + 457700*x^12 +...)*x^3/3

%e + (1 + 35*x^4 + 3830*x^8 + 570451*x^12 + 98118690*x^16 +...)*x^4/4

%e + (1 + 126*x^5 + 54127*x^10 + 32006130*x^15 +...)*x^5/5

%e + (1 + 462*x^6 + 782761*x^12 + 1841287756*x^18 +...)*x^6/6 +...

%e + exp( Sum_{k>=1} binomial(2*n*k,n*k)/2*x^(n*k)/k )*x^n/n +...

%e Explicitly,

%e log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 27*x^4/4 + 71*x^5/5 + 339*x^6/6 + 925*x^7/7 + 4347*x^8/8 + 13714*x^9/9 +...

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

%Y Cf. A205503, A000108.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 27 2012