A089468 Hyperbinomial transform of A089467 and also the 2nd hyperbinomial transform of A089466.
1, 3, 15, 110, 1083, 13482, 203569, 3618540, 74058105, 1715620148, 44384718879, 1268498827752, 39692276983555, 1349678904881400, 49556966130059553, 1954156038072106448, 82363978221026323761, 3695194039210436996400
Offset: 0
Keywords
Programs
-
Mathematica
CoefficientList[Series[(LambertW[-x]^2*E^(-1/2*LambertW[-x]^2))/(x^2*(1+LambertW[-x])), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jul 09 2013 *) -
PARI
a(n)=if(n<0,0,sum(m=0,n,sum(j=0,m,binomial(m,j)*binomial(n,n-m-j)*(n+1)^(n-m-j)*(m+j)!/(-2)^j)/m!))
Formula
a(n) = Sum_{k=0..n} (n-k+1)^(n-k-1)*C(n, k)*A089467(k).
a(n) = Sum_{k=0..n} 2*(n-k+2)^(n-k-1)*C(n, k)*A089466(k).
a(n) = Sum_{m=0..n} (Sum_{j=0..m} C(m, j)*C(n, n-m-j)*(n+1)^(n-m-j)*(m+j)!/(-2)^j)/m!.
E.g.f.: (LambertW(-x)^2*exp(-1/2*LambertW(-x)^2))/(x^2*(1+LambertW(-x))). - Vladeta Jovovic, Oct 26 2004
a(n) ~ exp(3/2)*n^n. - Vaclav Kotesovec, Jul 09 2013
Comments