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%I #49 May 16 2018 20:00:46
%S 1,1,34,2971,513559,149670844,66653198353,42429389528215,
%T 36788942253042556,41888564490333642283,60862147523250910055785,
%U 110264570238241604072673394,244397290937585028603794094349,652229940568729289038518033117685,2067551365133160531453420400711013314,7694635622932764203876848262780670955447
%N Generalized Bell numbers.
%C a(n) occurs in the process of normal ordering of the n-th power of a product of the cubes of the boson creation and boson annihilation operators.
%C a(11) = 110264570238241604072673394 =~ 10^26.
%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_{3}(x)), where [x^n] denotes the coefficient of x^n in the Taylor series for B_{3}(x).
%C a(n) is row 3 of the square array representation of A090210. (End)
%H P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://arXiv.org/abs/quant-ph/0212072">The Boson Normal Ordering Problem and Generalized Bell Numbers</a>, arXiv:quant-ph/0212072, 2002.
%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="http://dx.doi.org/10.1016/S0375-9601(03)00194-4">The general boson normal ordering problem</a>, Phys. Lett. A 309 (2003) 198-205.
%H P. Codara, O. M. D'Antona, P. Hell, <a href="http://arxiv.org/abs/1308.1700">A simple combinatorial interpretation of certain generalized Bell and Stirling numbers</a>, arXiv preprint arXiv:1308.1700 [cs.DM], 2013.
%H P. Codara, O. M. D’Antona, P. Hell, <a href="http://dx.doi.org/10.1016/j.disc.2013.11.010">A simple combinatorial interpretation of certain generalized Bell and Stirling numbers</a>, Discrete Math. 318 (2014), 53--57. MR3141626
%H S.-M. Ma, T. Mansour, M. Schork. <a href="http://arxiv.org/abs/1308.0169">Normal ordering problem and the extensions of the Stirling grammar</a>, arXiv preprint arXiv:1308.0169 [math.CO], 2013.
%H Toufik Mansour, Matthias Schork and Mark Shattuck, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Schork/schork2.html">The Generalized Stirling and Bell Numbers Revisited</a>, Journal of Integer Sequences, Vol. 15 (2012), #12.8.3.
%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. Riedel, <a href="http://math.stackexchange.com/questions/2165093/">Set partitions of unique elements from an n-by-m matrix where elements from the same row may not be in the same partition</a>
%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)= exp(-1) * Sum_{k>=0} ((k+3)!)^n/((k+3)!*(k!)^n), n>=1. This is a Dobinski-type summation formula.
%F a(n)= exp(-1) * Sum_{k>=3} (k*(k-1)*(k-2))^n)/k!, n>=1. Usually a(0) := 1. (From eq.(26) with r=3 of the Schork reference; rewritten original eq.(25) with r=3 of the Blasiak et al. reference.)
%F E.g.f. with a(0) := 1: (sum((exp(k*(k-1)*(k-2)*x))/k!, k=3..infinity)+5/2)/exp(1). From top of p. 4656 with r=3 of the Schork reference.
%p A069223 := proc(n) local r,s,i;
%p if n=0 then 1 else r := [seq(4,i=1..n-1)]; s := [seq(1,i=1..n-1)];
%p exp(-x)*6^(n-1)*hypergeom(r,s,x); round(evalf(subs(x=1,%),99)) fi end:
%p seq(A069223(n),n=1..15); # _Peter Luschny_, Mar 30 2011
%t f[n_] := f[n] = Sum[(k + 3)!^n/((k + 3)!*(k!^n)*E), {k, 0, Infinity}]; Table[ f[n], {n, 1, 9}]
%t a[n_] := (* row sum of A078741 *) Sum[(-1)^k*Sum[(-1)^p*((p - 2)*(p - 1)*p)^n*Binomial[k, p], {p, 3, k}]/k!, {k, 3, 3n}]; Array[a, 15] (* _Jean-François Alcover_, Sep 01 2015 *)
%o (PARI) default(realprecision, 500); for(n=0, 20, print1(if(n==0, 1, round(exp(-1)*sum(k=0, 500, ((k+3)!)^n/( (k+3)!*(k!)^n)))), ", ")) \\ _G. C. Greubel_, May 15 2018
%Y Cf. A000110 and A020556, if k+3 is replaced by k+1 or k+2, respectively.
%Y Cf. A090210.
%K nonn,easy
%O 0,3
%A _Karol A. Penson_, Apr 12 2002
%E Edited by _Robert G. Wilson v_, Apr 30 2002
%E a(0)=1 prepended by _Alois P. Heinz_, Aug 01 2016
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