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A153278 Array read by antidiagonals of higher order Fubini numbers. 2
1, 1, 3, 1, 4, 13, 1, 5, 23, 75, 1, 6, 36, 175, 541, 1, 7, 52, 342, 1662, 4683, 1, 8, 71, 594, 4048, 18937, 47293, 1, 9, 93, 949, 8444, 57437, 251729, 545835, 1, 10, 118, 1425, 15775, 143783, 950512, 3824282, 7087261, 1, 11, 146, 2040, 27146, 313920, 2854261 (list; table; graph; refs; listen; history; text; internal format)
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.

REFERENCES

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

LINKS

Alois P. Heinz, Antidiagonals n = 1..101, flattened

Istvan Mezo, On powers of Stirling matrices, Dec 21, 2008.

EXAMPLE

The table on p.6 of Mezo begins:

===========================================================

F_p,n|n=1|n=2|n=3.|.n=4.|..n=5.|....n=6.|.....n=7.|comment

===========================================================

p=1..|.1.|.3.|.13.|..75.|..541.|...4683.|...47293.|.A000670

p=2..|.1.|.4.|.23.|.175.|.1662.|..18937.|..251729.|.A083355

p=3..|.1.|.5.|.36.|.342.|.4048.|..57437.|..950512.|.A099391

p=4..|.1.|.6.|.52.|.594.|.8444.|.143783.|.2854261.|.new

p=5..|.1.|.7.|.71.|.949.|15775.|.313920.|.7279795.|.new

===========================================================

MAPLE

with (combinat): f:= proc(n) option remember; local k; if n<=1 then 1 else add (binomial (n, k) *f(n-k), k=1..n) fi end: stirtr:= proc(a) proc (n) option remember; add ( a(k) *stirling2(n, k), k=0..n) end end: F:= (p, n)-> (stirtr@@(p-1)) (f)(n): seq (seq (F (d-n, n), n=1..d-1), d=1..13); # Alois P. Heinz, Feb 02 2009

CROSSREFS

Cf. A000670, A083355, A099391, A153277.

Sequence in context: A010756 A191857 A248044 * A010284 A095328 A066712

Adjacent sequences:  A153275 A153276 A153277 * A153279 A153280 A153281

KEYWORD

easy,nonn,tabl

AUTHOR

Jonathan Vos Post, Dec 22 2008

EXTENSIONS

More terms from Alois P. Heinz, Feb 02 2009

STATUS

approved

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Last modified November 26 00:25 EST 2014. Contains 250017 sequences.