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

%I #6 Mar 30 2012 18:37:34

%S 1,1,3,8,27,79,292,900,3369,11131,41742,139002,546529,1829265,7113275,

%T 24903332,96838366,335955634,1345392796,4673507879,18615675149,

%U 66574809640,262503701044,933024442958,3796054662682,13418892324198,53794826244366,195768193280117

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

%F G.f.: exp( Sum_{n>=1} C_n(x^n)^2 * 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 + 3*x^2 + 8*x^3 + 27*x^4 + 79*x^5 + 292*x^6 + 900*x^7 +...

%e By definition:

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

%e + (1 + 6*x^2 + 53*x^4 + 554*x^6 + 6362*x^8 + 77580*x^10 +...)*x^2/2

%e + (1 + 20*x^3 + 662*x^6 + 26780*x^9 + 1205961*x^12 +...)*x^3/3

%e + (1 + 70*x^4 + 8885*x^8 + 1409002*x^12 + 250837850*x^16 +...)*x^4/4

%e + (1 + 252*x^5 + 124130*x^10 + 77652264*x^15 +...)*x^5/5

%e + (1 + 924*x^6 + 1778966*x^12 + 4405846676*x^18 +...)*x^6/6 +...

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

%e Explicitly,

%e log(A(x)) = x + 5*x^2/2 + 16*x^3/3 + 69*x^4/4 + 211*x^5/5 + 992*x^6/6 + 3004*x^7/7 + 13797*x^8/8 + 45745*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)*x^(m*k)/k)+x*O(x^n)))),n)}

%Y Cf. A205504, A000108.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 27 2012