A153277 Array read by antidiagonals of higher order Bell numbers.

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, 1304, 2471, 877, 1, 8, 70, 561, 3455, 12915, 19302, 4140, 1, 9, 92, 910, 7556, 44590, 146115, 167894, 21147, 1, 10, 117, 1380, 14532, 120196, 660665, 1855570, 1606137, 115975

Offset

1,3

Comments

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.

Links

Table of n, a(n) for n=1..55.

E. T. Bell, The iterated exponential integers, Ann. Math. 39(3) (1938), 539-557.

J. Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353.

Istvan Mezo, On powers of Stirling matrices, arXiv:0812.4047.

K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela, A. I. Solomon, Hierarchical Dobinski-type relations via substitution and the moment problem, J.Phys. A: Math.Gen. 37 3475-3487 (2004).

Example

The table on p.4 of Mezo begins:
===========================================================
B_p,n|n=1|n=2|n=3.|.n=4.|..n=5.|....n=6.|.....n=7.|comment
===========================================================
p=1..|.1.|.2.|..5.|..15.|...52.|....203.|.....877.|.A000110
p=2..|.1.|.3.|.12.|..60.|..358.|...2471.|...19302.|.A000258
p=3..|.1.|.4.|.22.|.154.|.1304.|..12915.|..146115.|.A000307
p=4..|.1.|.5.|.35.|.315.|.3455.|..44590.|..660665.|.A000357
p=5..|.1.|.6.|.51.|.561.|.7556.|.120196.|.2201856.|.A000405
===========================================================

Maple

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):
seq(seq(B(d-n, n), n=1..d-1), d=1..12); # Alois P. Heinz, Feb 02 2009

Mathematica

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 *)

Crossrefs

Cf. A000110, A000258, A000307, A000357, A000405, A111672.

From Alois P. Heinz, Feb 02 2009: (Start)

Truncated and reflected version of A144150.

Cf. A001669, A081624, A081629, A081697, A081740, A000326, A005945. (End)

Sequence in context: A134247 A210225 A180906 * A104029 A208752 A119308

Adjacent sequences: A153274 A153275 A153276 * A153278 A153279 A153280

Keyword

easy,nonn,tabl

Author

Jonathan Vos Post, Dec 22 2008

Extensions

More terms from Alois P. Heinz, Feb 02 2009

Status

approved