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A153277 Array read by antidiagonals of higher order Bell numbers.

%I A153277 #16 Jan 28 2015 04:54:11
%S A153277 1,1,2,1,3,5,1,4,12,15,1,5,22,60,52,1,6,35,154,358,203,1,7,51,315,
%T A153277 1304,2471,877,1,8,70,561,3455,12915,19302,4140,1,9,92,910,7556,44590,
%U A153277 146115,167894,21147,1,10,117,1380,14532,120196,660665,1855570,1606137,115975
%N A153277 Array read by antidiagonals of higher order Bell numbers.
%C A153277 Mezo's abstract: The powers of matrices with Stirling number-coefficients are investigated. It is revealed that the elements of these matrices have a number of properties of the ordinary Stirling numbers. Moreover, "higher order" Bell, Fubini and Eulerian numbers can be defined. Hence we give a new interpretation for E. T. Bell's iterated exponential integers. In addition, it is worth to note that these numbers appear in combinatorial physics, in the problem of the normal ordering of quantum field theoretical operators.
%H A153277 E. T. Bell, <a href="http://www.jstor.org/stable/1968633">The iterated exponential integers</a>, Ann. Math. 39(3) (1938), 539-557.
%H A153277 J. Ginsburg, <a href="http://www.jstor.org/stable/4145040">Iterated exponentials</a>, Scripta Math., 11 (1945), 340-353.
%H A153277 Istvan Mezo, <a href="http://arxiv.org/abs/0812.4047">On powers of Stirling matrices</a>, arXiv:0812.4047.
%H A153277 K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela, A. I. Solomon, <a href="http://dx.doi.org/10.1088/0305-4470/37/10/011">Hierarchical Dobinski-type relations via substitution and the moment problem</a>, J.Phys. A: Math.Gen. 37 3475-3487 (2004).
%e A153277 The table on p.4 of Mezo begins:
%e A153277 ===========================================================
%e A153277 B_p,n|n=1|n=2|n=3.|.n=4.|..n=5.|....n=6.|.....n=7.|comment
%e A153277 ===========================================================
%e A153277 p=1..|.1.|.2.|..5.|..15.|...52.|....203.|.....877.|.A000110
%e A153277 p=2..|.1.|.3.|.12.|..60.|..358.|...2471.|...19302.|.A000258
%e A153277 p=3..|.1.|.4.|.22.|.154.|.1304.|..12915.|..146115.|.A000307
%e A153277 p=4..|.1.|.5.|.35.|.315.|.3455.|..44590.|..660665.|.A000357
%e A153277 p=5..|.1.|.6.|.51.|.561.|.7556.|.120196.|.2201856.|.A000405
%e A153277 ===========================================================
%p A153277 g:= proc(a) local b; b:=proc(n) option remember; if n=0 then 1 else (n-1)! *add (a(k)* b(n-k)/ (k-1)!/ (n-k)!, k=1..n) fi end end: B:= (p,n)-> (g@@p)(1)(n):
%p A153277 seq(seq(B(d-n, n), n=1..d-1), d=1..12); # _Alois P. Heinz_, Feb 02 2009
%t A153277 g[k_] := g[k] = Nest[Function[x, E^x-1], x, k]; a[n_, k_] := SeriesCoefficient[ 1+g[k+1], {x, 0, n}]*n!; Table[a[n, k-n+1], {k, 1, 12}, {n, 1, k}] // Flatten (* _Jean-François Alcover_, Jan 28 2015 *)
%Y A153277 Cf. A000110, A000258, A000307, A000357, A000405, A111672.
%Y A153277 From _Alois P. Heinz_, Feb 02 2009: (Start)
%Y A153277 Truncated and reflected version of A144150.
%Y A153277 Cf. A001669, A081624, A081629, A081697, A081740, A000326, A005945. (End)
%K A153277 easy,nonn,tabl
%O A153277 1,3
%A A153277 _Jonathan Vos Post_, Dec 22 2008
%E A153277 More terms from _Alois P. Heinz_, Feb 02 2009