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A153278
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Array read by antidiagonals of higher order Fubini numbers.
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2
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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
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OFFSET
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1,3
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COMMENTS
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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.
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REFERENCES
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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).
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LINKS
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Alois P. Heinz, Antidiagonals n = 1..101, flattened
Istvan Mezo, On powers of Stirling matrices, Dec 21, 2008.
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EXAMPLE
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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
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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
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MAPLE
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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
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CROSSREFS
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Cf. A000670, A083355, A099391, A153277.
Sequence in context: A065253 A010756 A191857 * A010284 A095328 A066712
Adjacent sequences: A153275 A153276 A153277 * A153279 A153280 A153281
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KEYWORD
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easy,nonn,tabl
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AUTHOR
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Jonathan Vos Post, Dec 22 2008
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EXTENSIONS
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More terms from Alois P. Heinz, Feb 02 2009
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STATUS
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approved
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