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A187783 De Bruijn's triangle, T(m,n) = (m*n)!/(n!^m) read by downward antidiagonals. 14
1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 6, 1, 1, 1, 20, 90, 24, 1, 1, 1, 70, 1680, 2520, 120, 1, 1, 1, 252, 34650, 369600, 113400, 720, 1, 1, 1, 924, 756756, 63063000, 168168000, 7484400, 5040, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

From Tilman Piesk, Oct 28 2014: (Start)

Number of permutations of a multiset that contains m different elements n times. These multisets have the signatures A249543(m,n-1) for m>=1 and n>=2.

In an m-dimensional Pascal tensor (the generalization of a symmetric Pascal matrix) P(x1,...,xn) = (x1+...+xn)!/(x1!*...*xn!), so the main diagonal of an m-dimensional Pascal tensor is D(n) = (m*n)!/(n!^m). These diagonals are the rows of this array (with m>0), which begins like this:

  n   0    1        2             3                 4                     5

m

0     1    1        1             1                 1                     1  A000012

1     1    1        1             1                 1                     1  A000012

2     1    2        6            20                70                   252  A000984

3     1    6       90          1680             34650                756756  A006480

4     1   24     2520        369600          63063000           11732745024  A008977

5     1  120   113400     168168000      305540235000       623360743125120  A008978

6     1  720  7484400  137225088000  3246670537110000  88832646059788350720  A008979

     A000142  A000680       A014606           A014608               A014609

A089759 is the transpose of this matrix. A034841 is its diagonal. A141906 is its lower triangle. A120666 is the upper triangle of this matrix with indices starting from 1. A248827 are the diagonal sums (or the row sums of the triangle).

(End)

LINKS

Tilman Piesk, First 54 rows of the triangle, flattened

T. Chappell, A. Lascoux, S. Ole Warnaar, W. Zudilin, Logarithmic and complex constant term identities, arXiv:1112.3130 [math.CO], 2012.

Tilman Piesk, Array for indices 0..16

Tilman Piesk, PHP code used to create the b-file

Tilman Piesk, Illustration of the multisets for m,n=0..4

Wikipedia, Permutations of multisets, Pascal matrix and simplex

FORMULA

T(m,n) = (m*n)!/(n!^m).

EXAMPLE

T(3,5) = (3*5)!/(5!^3) = 756756 = A014609(3) = A006480(5) is the number of permutations of a multiset that contains 3 different elements 5 times, e.g., {1,1,1,1,1,2,2,2,2,2,3,3,3,3,3}.

MATHEMATICA

T[n_, k_] := (k*n)!/(n!)^k; Table[T[n, k - n], {k, 9}, {n, 0, k - 1}] // Flatten

CROSSREFS

Cf. A141906, A120666.

Rows: A000012, A000984, A006480, A006480, A008978, A008979, A071549, A071550, A071551, A071552

Columns: A000012, A000142, A000680, A014606, A014608, A014609, A248814, A172603, A172609, A172613

Another version is A089759. Diagonal gives: A034841. - Alois P. Heinz, Jan 23 2013

Row sums of the triangle: A248827

Sequence in context: A225816 A227655 A064992 * A089759 A088152 A049270

Adjacent sequences:  A187780 A187781 A187782 * A187784 A187785 A187786

KEYWORD

nonn,tabl,easy

AUTHOR

Robert G. Wilson v, Jan 05 2013

EXTENSIONS

Row m=0 prepended by Tilman Piesk, Oct 28 2014

STATUS

approved

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Last modified October 18 21:22 EDT 2018. Contains 316326 sequences. (Running on oeis4.)