A121630 Finite sum involving signless Stirling numbers of the first kind and the Bell numbers. Appears in the process of normal ordering of n-th power of (a)^3*(a+*a), where a+ and a are boson creation and annihilation operators, respectively.

1, 4, 29, 302, 4089, 68056, 1342949, 30635074, 792915057, 22952573484, 734630159341, 25757268041814, 981687991859689, 40407710444419072, 1786311057929722549, 84404172618241446506, 4244839086310722228449

Offset: 0

Author

Karol A. Penson, Aug 12 2006

Keywords

• nonn

Crossrefs

Cf. A002720, A121629, A121631, A239301.

Programs

Mathematica

CoefficientList[Series[E^(((1-3*x)^(-1/3))-1)/(1-3*x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Mar 14 2014 *)

Formula

a(n)=sum(abs(stirling1(n+1,p))*3^(n-p+1)*bell(p-1),p=1..n+1), n=0,1....

E.g.f.: exp(((1-3*x)^(-1/3))-1)/(1-3*x). - Vladeta Jovovic, Aug 13 2006

Recurrence: a(n) = 3*(4*n - 5)*a(n-1) - (54*n^2 - 189*n + 173)*a(n-2) + (108*n^3 - 729*n^2 + 1659*n - 1271)*a(n-3) - 9*(n-3)^2*(3*n - 8)*(3*n - 7)*a(n-4). - Vaclav Kotesovec, Mar 14 2014

a(n) ~ 1/2 * 3^(n+7/8) * exp(4*n^(1/4)/3^(3/4) - n - 1) * n^(n+3/8). - Vaclav Kotesovec, Mar 14 2014