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

0,2

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)

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

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 *)

Prog

(PARI) x='x+O('x^30); Vec(serlaplace(exp(((1-2*x)^(-1/2))-1)/(1-2*x))) \\ G. C. Greubel, May 17 2018
(Magma) m:=25; R<x>:=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

Crossrefs

Cf. A002720, A121630, A121631, A239301.

Sequence in context: A145158 A132070 A362204 * A351218 A200793 A141625

Adjacent sequences: A121626 A121627 A121628 * A121630 A121631 A121632

Keyword

nonn

Author

Karol A. Penson, Aug 12 2006

Extensions

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

Status

approved