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a(n) = (1/e) * Sum_{k>=0} ((k+4)!/k!)^(n-1)/k!.
8

%I #43 Feb 02 2019 02:32:52

%S 1,1,209,163121,326922081,1346634725665,9939316337679281,

%T 119802044788535500753,2205421644124274191535553,

%U 58945667435045762187763602753,2198513228897522394476415669503377,110833342180980170285766876408530089329

%N a(n) = (1/e) * Sum_{k>=0} ((k+4)!/k!)^(n-1)/k!.

%C This is a Dobinski-type summation formula.

%C Terms quickly become gigantic: a(15) = 9142140479823239889945170786704021785456107245847570873873. a(n) appears in the process of ordering the n-th power of a product of fourth power of boson creation and fourth power of boson annihilation operators.

%C From _Peter Luschny_, Mar 27 2011: (Start)

%C Let B_{m}(x) = Sum_{j>=0} (exp(j!/(j-m)!*x-1)/j!) then a(n) = n! [x^n] Taylor(B_{4}(x)), where [x^n] denotes the coefficient of x^k in the Taylor series for B_{4}(x).

%C a(n) is row 4 of the square array representation of A090210. (End)

%H P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://www.arXiv.org/abs/quant-ph/0402027">The general boson normal ordering problem</a>, arXiv:quant-ph/0402027, 2004.

%H P. Blasiak, K. A. Penson and A. I. Solomon, <a href="https://doi.org/10.1016/S0375-9601(03)00194-4">The general boson normal ordering problem/a>, Phys. Lett. A 309 (2003) 198-205.

%H K. A. Penson, P. Blasiak, A. Horzela, A. I. Solomon and G. H. E. Duchamp, <a href="http://arxiv.org/abs/0904.0369">Laguerre-type derivatives: Dobinski relations and combinatorial identities</a>, J. Math. Phys. 50, 083512 (2009).

%H M. Schork, <a href="http://dx.doi.org/10.1088/0305-4470/36/16/314">On the combinatorics of normal ordering bosonic operators and deforming it</a>, J. Phys. A 36 (2003) 4651-4665.

%F a(n) = (1/e)*Sum_{k>=4} fallfac(k, 4)^n / k!, n >= 1, with fallfac(n, m) := A008279(n, m) (falling factorials). (From eq.(26) with r=4 of the Schork reference.)

%F E.g.f. with a(0) := 1: (1/e)*(Sum_{k>=4} e^(fallfac(k, 4)*x)/k! + 8/3). From top of p. 4656 with r=4 of the Schork reference.

%p A071379 := proc(n) local r,s,i;

%p if n=0 then 1 else r := [seq(5,i=1..n-1)]; s := [seq(1,i=1..n-1)];

%p exp(-x)*24^(n-1)*hypergeom(r,s,x); round(evalf(subs(x=1,%),99)) fi end:

%p seq(A071379(n),n=0..10); # _Peter Luschny_, Mar 30, 2011

%t a[n_] := Sum[FactorialPower[k, 4]^n/k!, {k, 4, Infinity}]/E; a[0] = 1; Array[a, 12, 0] (* _Jean-François Alcover_, Sep 01 2016 *)

%o (PARI) default(realprecision, 500); for(n=0, 20, print1(if(n==0, 1, round(exp(-1)*sum(k=0, 500, ((k+4)!/k!)^(n-1)/k!))), ", ")) \\ _G. C. Greubel_, May 15 2018

%Y Cf. A000110, A020556 and A069223, when k+4 is replaced by k+1, k+2 or k+3 respectively.

%Y Cf. A090210.

%K nonn

%O 0,3

%A _Karol A. Penson_, May 22 2002

%E a(0)=1 prepended by _Alois P. Heinz_, Aug 01 2016

%E If it is proved that A283153 and A071379 are the same, then the entries should be merged and A283153 recycled. - _N. J. A. Sloane_, Mar 06 2017