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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.)