A121629 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)^2*(a+*a), where a+ and a are boson creation and annihilation operators, respectively.

1, 3, 16, 121, 1179, 14026, 196783, 3177861, 58019356, 1181098459, 26515026561, 650572403218, 17316566815441, 496889918749251, 15288155067806104, 502024850361876481, 17522822345606176083, 647790109599863145106, 25283238154309049107231

Offset: 0

Author

Karol A. Penson, Aug 12 2006

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..350
• K. A. Penson, P. Blasiak, A. Horzela, G. H. E. Duchamp and A. I. Solomon, Laguerre-type derivatives: Dobinski relations and combinatorial identities, J. Math. Phys. vol. 50, 083512 (2009)

Crossrefs

Cf. A002720, A121630, A121631, A239301.

Programs

Magma

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(((1-2*x)^(-1/2))-1)/(1-2*x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 17 2018

Mathematica

CoefficientList[Series[E^(((1-2*x)^(-1/2))-1)/(1-2*x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 29 2013 *)

PARI

x='x+O('x^30); Vec(serlaplace(exp(((1-2*x)^(-1/2))-1)/(1-2*x))) \\ G. C. Greubel, May 17 2018

Formula

a(n) = Sum_{p=1..n+1} abs(stirling1(n+1,p))*2^(n-p+1)*bell(p-1), n=0,1...

E.g.f.: exp(((1-2*x)^(-1/2))-1)/(1-2*x). - Vladeta Jovovic, Aug 13 2006

Recurrence: a(n) = (6*n-5)*a(n-1) - (2*n-3)*(6*n-7)*a(n-2) + 4*(2*n-3)*(n-2)^2*a(n-3). - Vaclav Kotesovec, Jun 29 2013

a(n) ~ 2^(n+5/6)*exp(3/2*(2*n)^(1/3)-1-n)*n^(n+1/3)/sqrt(3). - Vaclav Kotesovec, Jun 29 2013

Extensions

Terms a(17) onward added by G. C. Greubel, May 17 2018