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A272644
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Triangle read by rows: T(n,m) = Sum_{i=0..m} Stirling2(m+1,i+1)*(-1)^(m-i)*i^(n-m)*i!, for n >= 2, m = 1..n-1.
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3
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1, 1, 1, 1, 5, 1, 1, 13, 13, 1, 1, 29, 73, 29, 1, 1, 61, 301, 301, 61, 1, 1, 125, 1081, 2069, 1081, 125, 1, 1, 253, 3613, 11581, 11581, 3613, 253, 1, 1, 509, 11593, 57749, 95401, 57749, 11593, 509, 1, 1, 1021, 36301, 268381, 673261, 673261, 268381, 36301, 1021, 1
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OFFSET
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2,5
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COMMENTS
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Gives number of bitriangular permutations. Could be prefixed with an initial row containing a single 1. - N. J. A. Sloane, Jan 10 2018
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LINKS
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Gheorghe Coserea, Rows n = 2..101, flattened
F. Alayont and N. Krzywonos, Rook Polynomials in Three and Higher Dimensions, 2012.
Beáta Bényi, A Bijection for the Boolean Numbers of Ferrers Graphs, Graphs and Combinatorics (2022) Vol. 38, No. 10.
Beata Bényi and Peter Hajnal, Combinatorial properties of poly-Bernoulli relatives, arXiv preprint arXiv:1602.08684 [math.CO], 2016. See D_{n,k}.
Irving Kaplansky and John Riordan, The problem of the rooks and its applications, Duke Mathematical Journal 13.2 (1946): 259-268. The array is on page 267.
Irving Kaplansky and John Riordan, The problem of the rooks and its applications, in Combinatorics, Duke Mathematical Journal, 13.2 (1946): 259-268. [Annotated scanned copy]
J. Riordan, Letter to N. J. A. Sloane, Dec. 1976
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FORMULA
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T(n,m) = Sum_{i=0..m} Stirling2(m+1, i+1)*(-1)^(m-i)*i^(n-m)*i!, for n>=2, m=1..n-1, where Stirling2(n,k) is defined by A008277.
A001469(n+1) = Sum_{m=1..2*n-1} (-1)^(m-1)*T(2*n,m). - Gheorghe Coserea, May 18 2016
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EXAMPLE
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Triangle begins:
n\m [1] [2] [3] [4] [5] [6] [7] [8]
[2] 1;
[3] 1, 1;
[4] 1, 5, 1;
[5] 1, 13, 13, 1;
[6] 1, 29, 73, 29, 1;
[7] 1, 61, 301, 301, 61, 1;
[8] 1, 125, 1081, 2069, 1081, 125, 1;
[9] 1, 253, 3613, 11581, 11581, 3613, 253, 1;
...
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MAPLE
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A272644 := proc(n, m)
add(combinat[stirling2](m+1, i+1)*(-1)^(m-i)*i^(n-m)*i!, i=0..m) ;
end proc:
seq(seq(A272644(n, m), m=1..n-1), n=2..10) ; # R. J. Mathar, Mar 04 2018
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MATHEMATICA
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Table[Sum[StirlingS2[m + 1, i + 1] (-1)^(m - i) i^(n - m) i!, {i, 0, m} ], {n, 11}, {m, n - 1}] /. {} -> {0} // Flatten (* Michael De Vlieger, May 19 2016 *)
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PROG
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(PARI)
A(n, m) = sum(i=0, m, stirling(m+1, i+1, 2) * (-1)^((m-i)%2) * i^(n - m) * i!);
concat(vector(10, n, vector(n, m, A(n+1, m)))) \\ Gheorghe Coserea, May 16 2016
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CROSSREFS
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Column 2 is A036563.
Largest term in each row gives A272645.
Second diagonal from the right is 2^i - 3.
Third diagonal from the right edge is A006230.
For row sums see A297195.
Cf. A008277, A001469.
Sequence in context: A143007 A152654 A176487 * A157177 A298240 A299366
Adjacent sequences: A272641 A272642 A272643 * A272645 A272646 A272647
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KEYWORD
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nonn,tabl
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AUTHOR
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N. J. A. Sloane, May 07 2016
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EXTENSIONS
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More terms from Gheorghe Coserea, May 16 2016
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STATUS
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approved
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