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 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. 3
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 2,5 COMMENTS Gives number of bitriangular permutations. Could be prefixed with an initial row containing a single 1. - N. J. A. Sloane, Jan 10 2018 LINKS Gheorghe Coserea, Rows n = 2..101, flattened F. Alayont and N. Krzywonos, Rook Polynomials in Three and Higher Dimensions, 2012. Beata Bényi, 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 FORMULA 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 EXAMPLE 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; ... MAPLE 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 MATHEMATICA 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 *) PROG (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 CROSSREFS 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 KEYWORD nonn,tabl AUTHOR N. J. A. Sloane, May 07 2016 EXTENSIONS More terms from Gheorghe Coserea, May 16 2016 STATUS approved

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Last modified May 25 17:53 EDT 2020. Contains 334595 sequences. (Running on oeis4.)