A231274 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (k + x) / (1 - k*x - x^2).

1, 1, 4, 18, 104, 736, 6232, 61632, 698144, 8917120, 126807520, 1987075872, 34018221728, 631698903712, 12645901972000, 271482140140704, 6221487421328672, 151587364647728032, 3912949321334320672, 106670353381399285920, 3062317963564624162592, 92345208262957730327968

Offset

0,3

Comments

Compare to the identity:

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

Compare also to the identity:

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

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..268

Formula

a(n) ~ n! / (2 * (log(2))^(n+1)). - Vaclav Kotesovec, Oct 31 2014

Example

G.f.: A(x) = 1 + x + 4*x^2 + 18*x^3 + 104*x^4 + 736*x^5 + 6232*x^6 +...
where
A(x) = 1 + x*(1+x)/(1-x-x^2) + x^2*(1+x)*(2+x)/((1-x-x^2)*(1-2*x-x^2)) + x^3*(1+x)*(2+x)*(3+x)/((1-x-x^2)*(1-2*x-x^2)*(1-3*x-x^2)) + x^4*(1+x)*(2+x)*(3+x)*(4+x)/((1-x-x^2)*(1-2*x-x^2)*(1-3*x-x^2)*(1-4*x-x^2)) +...

Prog

(PARI) {a(n)=polcoeff( sum(m=0, n, x^m*prod(k=1, m, (k+x)/(1-k*x-x^2 +x*O(x^n))) ), n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A231352, A231291.

Sequence in context: A159666 A349021 A316186 * A051827 A007711 A321278

Adjacent sequences: A231271 A231272 A231273 * A231275 A231276 A231277

Keyword

nonn

Author

Paul D. Hanna, Nov 06 2013

Status

approved