This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A272644 #78 Oct 17 2024 21:45:04
%S A272644 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 A272644 1081,125,1,1,253,3613,11581,11581,3613,253,1,1,509,11593,57749,95401,
%U A272644 57749,11593,509,1,1,1021,36301,268381,673261,673261,268381,36301,1021,1
%N A272644 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 A272644 Gives number of bitriangular permutations. Could be prefixed with row 0 containing a single 1. - _N. J. A. Sloane_, Jan 10 2018
%H A272644 Gheorghe Coserea, <a href="/A272644/b272644.txt">Rows n = 2..101, flattened </a>
%H A272644 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 A272644 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 A272644 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 A272644 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 A272644 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 A272644 D. E. 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 A272644 D. E. Knuth, <a href="/A372066/a372066.txt">Notes on four arrays of numbers arising from the enumeration of CRC constraints and min-and-max-closed constraints</a>, May 06 2024. Mentions this sequence.
%H A272644 J. Riordan, <a href="/A006230/a006230.pdf">Letter to N. J. A. Sloane, Dec. 1976</a>.
%F A272644 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 A272644 A001469(n+1) = Sum_{m=1..2*n-1} (-1)^(m-1)*T(2*n,m). - _Gheorghe Coserea_, May 18 2016
%e A272644 Triangle begins:
%e A272644 n\m [1] [2] [3] [4] [5] [6] [7] [8]
%e A272644 [2] 1;
%e A272644 [3] 1, 1;
%e A272644 [4] 1, 5, 1;
%e A272644 [5] 1, 13, 13, 1;
%e A272644 [6] 1, 29, 73, 29, 1;
%e A272644 [7] 1, 61, 301, 301, 61, 1;
%e A272644 [8] 1, 125, 1081, 2069, 1081, 125, 1;
%e A272644 [9] 1, 253, 3613, 11581, 11581, 3613, 253, 1;
%e A272644 ...
%p A272644 A272644 := proc(n,m)
%p A272644 add(combinat[stirling2](m+1,i+1)*(-1)^(m-i)*i^(n-m)*i!,i=0..m) ;
%p A272644 end proc:
%p A272644 seq(seq(A272644(n,m),m=1..n-1),n=2..10) ; # _R. J. Mathar_, Mar 04 2018
%t A272644 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 A272644 (PARI)
%o A272644 A(n,m) = sum(i=0, m, stirling(m+1, i+1, 2) * (-1)^((m-i)%2) * i^(n - m) * i!);
%o A272644 concat(vector(10, n, vector(n, m, A(n+1, m)))) \\ _Gheorghe Coserea_, May 16 2016
%Y A272644 Column 2 is A036563.
%Y A272644 Largest term in each row gives A272645.
%Y A272644 Second diagonal from the right is 2^i - 3.
%Y A272644 Third diagonal from the right edge is A006230.
%Y A272644 T(2n,n) gives A048144.
%Y A272644 For row sums see A297195.
%Y A272644 Cf. A008277, A001469, A371761.
%K A272644 nonn,tabl
%O A272644 2,5
%A A272644 _N. J. A. Sloane_, May 07 2016
%E A272644 More terms from _Gheorghe Coserea_, May 16 2016