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

0,2

Links

Table of n, a(n) for n=0..16.

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

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 *)

Crossrefs

Cf. A002720, A121629, A121631, A239301.

Sequence in context: A348651 A360584 A127770 * A089470 A360834 A349599

Adjacent sequences: A121627 A121628 A121629 * A121631 A121632 A121633

Keyword

nonn

Author

Karol A. Penson, Aug 12 2006

Status

approved