A230682 O.g.f.: Sum_{n>=0} x^n * Product_{k=1..n} (k^2 + x)/(1 + k^2*x).

1, 1, 4, 21, 181, 2320, 41581, 991821, 30339364, 1156828681, 53761779721, 2990342767680, 196097039232121, 14969727522159481, 1315952342285654884, 131970189920614495581, 14974773731779775857021, 1908770813250950767227280, 271560466483540753565395621

Offset

0,3

Comments

Compare to an o.g.f. of Genocchi numbers of the first kind (A110501):

Sum_{n>=0} x^n * Product_{k=1..n} k^2/(1 + k^2*x).

Also, compare to a g.f. of Fibonacci numbers (A000045):

Sum_{n>=0} x^n * Product_{k=1..n} (k + x)/(1 + k*x).

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..230

Formula

a(n) ~ 2^(2*n+5) * n^(2*n+5/2) / (Pi^(2*n+3/2) * exp(2*n)). - Vaclav Kotesovec, Oct 28 2014

Example

G.f.: A(x) = 1 + x + 4*x^2 + 21*x^3 + 181*x^4 + 2320*x^5 + 41581*x^6 +...
where
A(x) = 1 + x*(1+x)/(1+x) + x^2*(1+x)*(4+x)/((1+x)*(1+4*x)) + x^3*(1+x)*(4+x)*(9+x)/((1+x)*(1+4*x)*(1+9*x)) + x^4*(1+x)*(4+x)*(9+x)*(16+x)/((1+x)*(1+4*x)*(1+9*x)*(1+16*x)) +...

Prog

(PARI) {a(n)=polcoeff(sum(m=0, n, x^m*prod(k=1, m, k^2+x+x*O(x^n))/prod(k=1, m, 1+k^2*x+x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A230740.

Sequence in context: A383379 A278993 A306067 * A231220 A231434 A393741

Adjacent sequences: A230679 A230680 A230681 * A230683 A230684 A230685

Keyword

nonn

Author

Paul D. Hanna, Oct 27 2013

Status

approved