%I #11 Jul 09 2013 03:46:33
%S 1,3,15,110,1083,13482,203569,3618540,74058105,1715620148,44384718879,
%T 1268498827752,39692276983555,1349678904881400,49556966130059553,
%U 1954156038072106448,82363978221026323761,3695194039210436996400
%N Hyperbinomial transform of A089467 and also the 2nd hyperbinomial transform of A089466.
%C See A088956 for the definition of the hyperbinomial transform.
%F 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!)).
%F E.g.f.: (LambertW(-x)^2*exp(-1/2*LambertW(-x)^2))/(x^2*(1+LambertW(-x))). - _Vladeta Jovovic_, Oct 26 2004
%F a(n) ~ exp(3/2)*n^n. - _Vaclav Kotesovec_, Jul 09 2013
%t 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 *)
%o (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!))
%Y Cf. A089466, A089467, A088956.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Nov 08 2003