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A099594
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Array read by antidiagonals: poly-Bernoulli numbers B(-k,n).
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16
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1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 8, 14, 8, 1, 1, 16, 46, 46, 16, 1, 1, 32, 146, 230, 146, 32, 1, 1, 64, 454, 1066, 1066, 454, 64, 1, 1, 128, 1394, 4718, 6902, 4718, 1394, 128, 1, 1, 256, 4246, 20266, 41506, 41506, 20266, 4246, 256, 1, 1, 512, 12866, 85310, 237686, 329462, 237686, 85310, 12866, 512, 1
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OFFSET
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0,5
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COMMENTS
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B_n^{(-k)} is the number of distinct n by k "lonesum matrices" where a matrix of entries 0 or 1 is called lonesum when it is uniquely reconstructible from its row and column sums. [Brewbaker]
B_n^{(-k)} is the cardinality of the set { sigma in S_{n+k}: -k <= i-sigma(i) <= n for all i=1,2,...,n+k }. [Launois]
T(n,k) is also the number of permutations on [n+k] in which each substring whose support belongs to {1, 2, ..., n} or {n+1, n+2, ..., n+k} is increasing. For example, with n = 2 and k = 3, the permutation 41532 does not qualify because the substring 53 has support in {n+1, n+2, ..., n+k} = {3,4,5} but is not increasing. T(2,1) = 4 counts 123, 132, 231, 312 while the permutations satisfying Launois' condition above are 123, 132, 213, 231. A bijection between these sets of permutations would be interesting. - David Callan, Jul 22 2008. (Corrected by Norman Do, Sep 01 2008)
T(n,k) is also the number of acyclic orientations of the complete bipartite graph K_{n,k}. - Vincent Pilaud, Sep 15 2020
When indexed as a triangular array, T(n,k) is the number of permutations of [n] in which 1 is in position k and the excedance entries are precisely the entries to the left of 1. See link. - David Callan, Dec 12 2021
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LINKS
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Beáta Bényi, Advances in Bijective Combinatorics, Ph. D. Dissertation, Doctoral School of Mathematics and Computer Science, University of Szeged, Bolyai Institute, 2014. See Table 1.
Masanobu Kaneko, Poly-Bernoulli numbers, Journal de théorie des nombres de Bordeaux, 9 no. 1 (1997), Pages 221-228.
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FORMULA
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pB(k, n) = (-1)^n * Sum[i=0..n, (-1)^i * i! * Stirling2(n, i) / (i+1)^k ].
E.g.f.: e^(x+y) / [e^x + e^y - e^(x+y)].
T(n, k) = Sum_{j=0..n} (j+1)^k*Sum_{i=0..j} (-1)^(n+j-i)*C(j, i)*(j-i)^n. - Paul D. Hanna, Nov 04 2004
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EXAMPLE
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1, 1, 1, 1, 1, 1, ...
1, 2, 4, 8, 16, 32, ...
1, 4, 14, 46, 146, 454, ...
1, 8, 46, 230, 1066, 4718, ...
1, 16, 146, 1066, 6902, 41506, ...
1, 32, 454, 4718, 41506, 329462, ...
...
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MAPLE
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A:= (n, k)-> add(Stirling2(n+1, i+1)*Stirling2(k+1, i+1)*
i!^2, i=0..min(n, k)):
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MATHEMATICA
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T[n_, k_] := Sum[(-1)^(j+n)*(1+j)^k*j!*StirlingS2[n, j], {j, 0, n}]; Table[ T[n-k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 30 2016 *)
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PROG
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(PARI) T(n, k)=sum(j=0, n, (j+1)^k*sum(i=0, j, (-1)^(n+j-i)*binomial(j, i)*(j-i)^n))
(PARI) T(n, k)=sum(j=0, min(n, k), j!^2*stirling(n+1, j+1, 2)*stirling(k+1, j+1, 2)); \\ Michel Marcus, Mar 05 2017
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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