A221095 E.g.f.: exp( Sum_{n>=1} q_binomial(2*n,n,x) * x^n/n ), where q_binomial(n,k,q) = Product_{j=1..n-k} (1-q^(j+k))/(1-q^j).

1, 1, 4, 15, 104, 750, 7254, 74214, 914528, 12202632, 183781080, 2974435200, 52965004872, 1006137926040, 20652503811744, 449786292039000, 10452618371303040, 256326394027746240, 6648055804021356864, 181094856954089764032, 5184169133931737988480

Offset

0,3

Comments

Compare to g.f. C(x) of the Catalan numbers, where C(x) = 1 + x*C(x)^2:

C(x)^2 = exp( Sum_{n>=1} binomial(2*n,n) * x^n/n ).

Links

Table of n, a(n) for n=0..20.

Formula

E.g.f.: exp( Sum_{n>=1} x^n/n * Product_{k=1..n} (1-x^(n+k))/(1-x^k) ).

Example

E.g.f.: A(x) = 1 + x + 4*x^2/2! + 15*x^3/3! + 104*x^4/4! + 750*x^5/5! +...
where
log(A(x)) = x*(1-x^2)/(1-x) + x^2*(1-x^3)*(1-x^4)/(2*(1-x)*(1-x^2)) + x^3*(1-x^4)*(1-x^5)*(1-x^6)/(3*(1-x)*(1-x^2)*(1-x^3)) + x^4*(1-x^5)*(1-x^6)*(1-x^7)*(1-x^8)/(4*(1-x)*(1-x^2)*(1-x^3)*(1-x^4)) +...

Prog

(PARI) {a(n)=n!*polcoeff(exp(sum(m=1, 31, x^m/m*prod(k=1, m, (1-x^(m+k))/(1-x^k)+x*O(x^n)))), n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A129528.

Sequence in context: A079128 A356524 A289489 * A339287 A081548 A207163

Adjacent sequences: A221092 A221093 A221094 * A221096 A221097 A221098

Keyword

nonn

Author

Paul D. Hanna, Jan 01 2013

Status

approved