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%I #15 Sep 08 2022 08:45:27
%S 1,3,16,121,1179,14026,196783,3177861,58019356,1181098459,26515026561,
%T 650572403218,17316566815441,496889918749251,15288155067806104,
%U 502024850361876481,17522822345606176083,647790109599863145106,25283238154309049107231
%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)^2*(a+*a), where a+ and a are boson creation and annihilation operators, respectively.
%H G. C. Greubel, <a href="/A121629/b121629.txt">Table of n, a(n) for n = 0..350</a>
%H K. A. Penson, P. Blasiak, A. Horzela, G. H. E. Duchamp and A. I. Solomon, <a href="http://arxiv.org/abs/0904.0369">Laguerre-type derivatives: Dobinski relations and combinatorial identities</a>, J. Math. Phys. vol. 50, 083512 (2009)
%F a(n) = Sum_{p=1..n+1} abs(stirling1(n+1,p))*2^(n-p+1)*bell(p-1), n=0,1...
%F E.g.f.: exp(((1-2*x)^(-1/2))-1)/(1-2*x). - _Vladeta Jovovic_, Aug 13 2006
%F 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
%F 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
%t CoefficientList[Series[E^(((1-2*x)^(-1/2))-1)/(1-2*x), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Jun 29 2013 *)
%o (PARI) x='x+O('x^30); Vec(serlaplace(exp(((1-2*x)^(-1/2))-1)/(1-2*x))) \\ _G. C. Greubel_, May 17 2018
%o (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
%Y Cf. A002720, A121630, A121631, A239301.
%K nonn
%O 0,2
%A _Karol A. Penson_, Aug 12 2006
%E Terms a(17) onward added by _G. C. Greubel_, May 17 2018
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