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%I #70 Apr 08 2024 19:03:40
%S 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,
%T 1081,125,1,1,253,3613,11581,11581,3613,253,1,1,509,11593,57749,95401,
%U 57749,11593,509,1,1,1021,36301,268381,673261,673261,268381,36301,1021,1
%N 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.
%C Gives number of bitriangular permutations. Could be prefixed with an initial row containing a single 1. - _N. J. A. Sloane_, Jan 10 2018
%H Gheorghe Coserea, <a href="/A272644/b272644.txt">Rows n = 2..101, flattened </a>
%H F. Alayont and N. Krzywonos, <a href="http://faculty.gvsu.edu/alayontf/notes/rook_polynomials_higher_dimensions_preprint.pdf">Rook Polynomials in Three and Higher Dimensions</a>, 2012.
%H Beáta Bényi, <a href="https://doi.org/10.1007/s00373-021-02442-2">A Bijection for the Boolean Numbers of Ferrers Graphs</a>, Graphs and Combinatorics (2022) Vol. 38, No. 10.
%H Beata Bényi and Peter Hajnal, <a href="https://arxiv.org/abs/1602.08684">Combinatorial properties of poly-Bernoulli relatives</a>, arXiv preprint arXiv:1602.08684 [math.CO], 2016. See D_{n,k}.
%H Irving Kaplansky and John Riordan, <a href="http://projecteuclid.org/euclid.dmj/1077473616">The problem of the rooks and its applications</a>, Duke Mathematical Journal 13.2 (1946): 259-268. The array is on page 267.
%H Irving Kaplansky and John Riordan, <a href="/A274105/a274105.pdf">The problem of the rooks and its applications</a>, in Combinatorics, Duke Mathematical Journal, 13.2 (1946): 259-268. [Annotated scanned copy]
%H Don Knuth, <a href="http://cs.stanford.edu/~knuth/papers/poly-Bernoulli.pdf">Parades and poly-Bernoulli bijections</a>, Mar 31 2024. See (16.2).
%H J. Riordan, <a href="/A006230/a006230.pdf">Letter to N. J. A. Sloane, Dec. 1976</a>.
%F 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.
%F A001469(n+1) = Sum_{m=1..2*n-1} (-1)^(m-1)*T(2*n,m). - _Gheorghe Coserea_, May 18 2016
%e Triangle begins:
%e n\m [1] [2] [3] [4] [5] [6] [7] [8]
%e [2] 1;
%e [3] 1, 1;
%e [4] 1, 5, 1;
%e [5] 1, 13, 13, 1;
%e [6] 1, 29, 73, 29, 1;
%e [7] 1, 61, 301, 301, 61, 1;
%e [8] 1, 125, 1081, 2069, 1081, 125, 1;
%e [9] 1, 253, 3613, 11581, 11581, 3613, 253, 1;
%e ...
%p A272644 := proc(n,m)
%p add(combinat[stirling2](m+1,i+1)*(-1)^(m-i)*i^(n-m)*i!,i=0..m) ;
%p end proc:
%p seq(seq(A272644(n,m),m=1..n-1),n=2..10) ; # _R. J. Mathar_, Mar 04 2018
%t 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 *)
%o (PARI)
%o A(n,m) = sum(i=0, m, stirling(m+1, i+1, 2) * (-1)^((m-i)%2) * i^(n - m) * i!);
%o concat(vector(10, n, vector(n, m, A(n+1, m)))) \\ _Gheorghe Coserea_, May 16 2016
%Y Column 2 is A036563.
%Y Largest term in each row gives A272645.
%Y Second diagonal from the right is 2^i - 3.
%Y Third diagonal from the right edge is A006230.
%Y For row sums see A297195.
%Y Cf. A008277, A001469, A371761.
%K nonn,tabl
%O 2,5
%A _N. J. A. Sloane_, May 07 2016
%E More terms from _Gheorghe Coserea_, May 16 2016
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