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A153277
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Array read by antidiagonals of higher order Bell numbers.
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3
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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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.
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REFERENCES
| E. T. Bell, The iterated exponential integers, Ann. Math. 39(3) (1938), 539-557.
J. Ginsburg, Iterated exponentials, Scripta Math. 11 (1945), 340-353.
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
| Istvan Mezo, On powers of Stirling matrices, arXiv:0812.4047.
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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
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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
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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); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Feb 02 2009]
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CROSSREFS
| Cf. A000110, A000258, A000307, A000357, A000405, A111672.
Contribution from Alois P. Heinz (heinz(AT)hs-heilbronn.de), Feb 02 2009: (Start)
Truncated and reflected version of A144150.
Cf. A001669, A081624, A081629, A081697, A081740, A000326, A005945. (End)
Sequence in context: A134081 A134247 A180906 * A104029 A119308 A110197
Adjacent sequences: A153274 A153275 A153276 * A153278 A153279 A153280
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| Jonathan Vos Post (jvospost3(AT)gmail.com), Dec 22 2008
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EXTENSIONS
| More terms from Alois P. Heinz (heinz(AT)hs-heilbronn.de), Feb 02 2009
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