A205503 G.f.: exp( Sum_{n>=1} x^n/n * exp( Sum_{k>=1} binomial(2*n*k,n*k)*x^(n*k)/k ) ).

1, 1, 3, 8, 27, 79, 292, 900, 3369, 11131, 41742, 139002, 546529, 1829265, 7113275, 24903332, 96838366, 335955634, 1345392796, 4673507879, 18615675149, 66574809640, 262503701044, 933024442958, 3796054662682, 13418892324198, 53794826244366, 195768193280117

Offset

0,3

Links

Table of n, a(n) for n=0..27.

Formula

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).

Example

 G.f.: A(x) = 1 + x + 3*x^2 + 8*x^3 + 27*x^4 + 79*x^5 + 292*x^6 + 900*x^7 +...
By definition:
log(A(x)) = (1 + 2*x + 5*x^2 + 14*x^3 + 42*x^4 + 132*x^5 +...)*x
+ (1 + 6*x^2 + 53*x^4 + 554*x^6 + 6362*x^8 + 77580*x^10 +...)*x^2/2
+ (1 + 20*x^3 + 662*x^6 + 26780*x^9 + 1205961*x^12 +...)*x^3/3
+ (1 + 70*x^4 + 8885*x^8 + 1409002*x^12 + 250837850*x^16 +...)*x^4/4
+ (1 + 252*x^5 + 124130*x^10 + 77652264*x^15 +...)*x^5/5
+ (1 + 924*x^6 + 1778966*x^12 + 4405846676*x^18 +...)*x^6/6 +...
+ exp( Sum_{k>=1} binomial(2*n*k,n*k)*x^(n*k)/k )*x^n/n +...
Explicitly,
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 +...

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)*x^(m*k)/k)+x*O(x^n)))), n)}

Crossrefs

Cf. A205504, A000108.

Sequence in context: A305049 A332833 A148823 * A242537 A323795 A148824

Adjacent sequences: A205500 A205501 A205502 * A205504 A205505 A205506

Keyword

nonn

Author

Paul D. Hanna, Jan 27 2012

Status

approved