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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. 4

%I

%S 1,4,29,302,4089,68056,1342949,30635074,792915057,22952573484,

%T 734630159341,25757268041814,981687991859689,40407710444419072,

%U 1786311057929722549,84404172618241446506,4244839086310722228449

%N 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.

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

%F E.g.f.: exp(((1-3*x)^(-1/3))-1)/(1-3*x). - _Vladeta Jovovic_, Aug 13 2006

%F 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

%F 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

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

%Y Cf. A002720, A121629, A121631, A239301.

%K nonn

%O 0,2

%A _Karol A. Penson_, Aug 12 2006

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Last modified August 22 05:46 EDT 2019. Contains 326172 sequences. (Running on oeis4.)