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A090210 Triangle of certain generalized Bell numbers.

%I A090210 #34 Jun 03 2024 22:50:23
%S A090210 1,1,1,2,1,1,5,7,1,1,15,87,34,1,1,52,1657,2971,209,1,1,203,43833,
%T A090210 513559,163121,1546,1,1,877,1515903,149670844,326922081,12962661,
%U A090210 13327,1,1,4140,65766991,66653198353,1346634725665,363303011071,1395857215,130922,1,1
%N A090210 Triangle of certain generalized Bell numbers.
%C A090210 Let B_{n}(x) = sum_{j>=0}(exp(j!/(j-n)!*x-1)/j!) and
%C A090210 S(n,k) = k! [x^k] taylor(B_{n}(x)), where [x^k] denotes the
%C A090210 coefficient of x^k in the Taylor series for B_{n}(x).
%C A090210 Then S(n,k) (n>0, k>=0) is the square array representation of the triangle.
%C A090210 To illustrate the cross-references of T(n,k) when written as a square array.
%C A090210 n\k         A000012,A000012,A002720,A069948,A182925,A182924,...
%C A090210 0: A000012: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%C A090210 1: A000110: 1, 1, 2, 5, 15, 52, 203, 877, 4140, ...
%C A090210 2: A020556: 1, 1, 7, 87, 1657, 43833, 1515903, ...
%C A090210 3: A069223: 1, 1, 34, 2971, 513559, 149670844, ...
%C A090210 4: A071379: 1, 1, 209, 163121, 326922081, ...
%C A090210 5: A090209: 1, 1, 1546, 12962661, 363303011071,...
%C A090210 6:  ...     1, 1, 13327, 1395857215, 637056434385865,...
%C A090210 Note that the sequence T(0,k) is not included in the data.
%C A090210 - _Peter Luschny_, Mar 27 2011
%D A090210 P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
%D A090210 M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.
%H A090210 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>
%H A090210 W. Lang, <a href="/A090210/a090210.txt">First 8 rows</a>.
%H A090210 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).
%F A090210 a(n, m) = Bell(m;n-(m-1)), n>= m-1 >=0, with Bell(m;k) := Sum_{p=m..m*k} S2(m;k, p), where S2(m;k, p) := (((-1)^p)/p!) * Sum_{r=m..p} ((-1)^r)*binomial(p, r)*fallfac(r, m)^k; with fallfac(n, m) := A008279(n, m) (falling factorials) and m<=p<=k*m, k>=1, m=1, 2, ..., else 0. From eqs.(6) with r=s->m and eq.(19) with S_{r, r}(n, k)-> S2(r;n, k) of the Blasiak et al. reference. [Corrected by _Sean A. Irvine_, Jun 03 2024]
%F A090210 a(n, m) = (Sum_{k>=m} fallfac(k, m)^(n-(m-1)))/exp(1), n>=m-1>=0, else 0. From eq.(26) with r->m of the Schork reference which is rewritten eq.(11) of the original Blasiak et al. reference.
%F A090210 E.g.f. m-th column (no leading zeros): (Sum_{k>=m} exp(fallfac(k, m)*x)/k!) + A000522(m)/m!)/exp(1). Rewritten from the top of p. 4656 of the Schork reference.
%e A090210 Triangle begins:
%e A090210 1;
%e A090210 1, 1;
%e A090210 2, 1, 1;
%e A090210 5, 7, 1, 1;
%e A090210 15, 87, 34, 1, 1;
%e A090210 52, 1657, 2971, 209, 1, 1;
%e A090210 203, 43833, 513559, 163121, 1546, 1, 1;
%p A090210 A090210_AsSquareArray := proc(n,k) local r,s,i;
%p A090210 if k=0 then 1 else r := [seq(n+1,i=1..k-1)]; s := [seq(1,i=1..k-1)];
%p A090210 exp(-x)*n!^(k-1)*hypergeom(r,s,x); round(evalf(subs(x=1,%),99)) fi end:
%p A090210 seq(lprint(seq(A090210_AsSquareArray(n,k),k=0..6)),n=0..6);
%p A090210 # _Peter Luschny_, Mar 30 2011
%t A090210 t[n_, k_] := t[n, k] = Sum[(n+j)!^(k-1)/(j!^k*E), {j, 0, Infinity}]; t[_, 0] = 1;
%t A090210 Flatten[ Table[ t[n-k+1, k], {n, 0, 8}, {k, n, 0, -1}]][[1 ;; 43]] (* _Jean-François Alcover_, Jun 17 2011 *)
%Y A090210 Cf. A000110, A020556, A069223, A071379, A090209, A002720, A069948, A182924, A182925, A182933.
%Y A090210 T(n,n) gives A070227.
%K A090210 nonn,easy,tabl
%O A090210 1,4
%A A090210 _Wolfdieter Lang_, Dec 01 2003