A205504 G.f.: exp( Sum_{n>=1} x^n/n * exp( Sum_{k>=1} binomial(2*n*k,n*k)/2 * x^(n*k)/k ) ).

1, 1, 2, 4, 11, 27, 92, 252, 906, 2787, 10191, 31594, 125998, 393021, 1535964, 5161328, 20221291, 66306664, 273756969, 897440988, 3664037417, 12621555612, 50496343297, 170909672792, 725703552284, 2427269270146, 9982179588261, 35179417316991, 143999051236064

Offset

0,3

Links

Table of n, a(n) for n=0..28.

Formula

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).

Example

 G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 11*x^4 + 27*x^5 + 92*x^6 +...
By definition:
log(A(x)) = (1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 +...)*x
+ (1 + 3*x^2 + 22*x^4 + 211*x^6 + 2306*x^8 + 27230*x^10 +...)*x^2/2
+ (1 + 10*x^3 + 281*x^6 + 10580*x^9 + 457700*x^12 +...)*x^3/3
+ (1 + 35*x^4 + 3830*x^8 + 570451*x^12 + 98118690*x^16 +...)*x^4/4
+ (1 + 126*x^5 + 54127*x^10 + 32006130*x^15 +...)*x^5/5
+ (1 + 462*x^6 + 782761*x^12 + 1841287756*x^18 +...)*x^6/6 +...
+ exp( Sum_{k>=1} binomial(2*n*k,n*k)/2*x^(n*k)/k )*x^n/n +...
Explicitly,
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 +...

Prog

 (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)}

Crossrefs

Cf. A205503, A000108.

Sequence in context: A234845 A148131 A234844 * A290337 A099412 A100094

Adjacent sequences: A205501 A205502 A205503 * A205505 A205506 A205507

Keyword

nonn

Author

Paul D. Hanna, Jan 27 2012

Status

approved