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A094385 Another version of triangular array in A062991 unsigned and transposed : triangle T(n,k), 0<=k<=n, read by rows; given by [0, 1, 1, 1, 1, 1, 1, ...] DELTA [1, 1, 1, 1, 1, 1, 1, 1, 1, ...] where DELTA is the operator defined in A084938. 5
1, 0, 1, 0, 1, 2, 0, 2, 6, 5, 0, 5, 20, 28, 14, 0, 14, 70, 135, 120, 42, 0, 42, 252, 616, 770, 495, 132, 0, 132, 924, 2730, 4368, 4004, 2002, 429, 0, 429, 3432, 11880, 23100, 27300, 19656, 8008, 1430, 0, 1430, 12870, 51051, 116688, 168300, 157080, 92820, 31824, 4862 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

Diagonals: A000007, A000108, 2*A001700; A000108, A002694. Row sums: A064092 (generalized Catalan C(2; n).

LINKS

Table of n, a(n) for n=0..54.

P. Barry, A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.4.

Yue Cai, Catherine Yan, Counting with Borel's triangle, Texas A&M University.

Yue Cai, Catherine Yan, Counting with Borel's triangle, arXiv:1804.01597 [math.CO], 2018.

B. Derrida, E. Domany and D. Mukamel, A exact solution of a one-dimensional asymmetric exclusion model with open boundaries, J. Stat. Phys. 69, 1992, 667-687; eqs.(20), (21), p. 672.

FORMULA

Sum_{k = 0..n} T(n, k)*x^(n-k) = C(x+1; n), generalized Catalan numbers; see left diagonals of triangle A064094: A000012, A000108, A064062..63, A064087..93 for x = -1, 0, ..., 9, respectively.

T(n,0)=0^n, T(n,k)=binomial(2*n,k-1)*binomial(2*n-k-1,n-k)/n for n>=1 and k>=1.

EXAMPLE

Triangle begins:

  1;

  0,   1;

  0,   1,    2;

  0,   2,    6,     5;

  0,   5,   20,    28,    14;

  0,  14,   70,   135,   120,    42;

  0,  42,  252,   616,   770,   495,   132;

  0, 132,  924,  2730,  4368,  4004,  2002,  429;

  0, 429, 3432, 11880, 23100, 27300, 19656, 8008, 1430; ...

MATHEMATICA

T[n_, k_] := Binomial[2n, k-1] Binomial[2n-k-1, n-k]/n; T[0, 0] = 1;

Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 19 2018 *)

CROSSREFS

See also A234950 for another version. - Philippe Deléham, Jan 11 2014

Sequence in context: A323845 A327116 A157491 * A291799 A295027 A225479

Adjacent sequences:  A094382 A094383 A094384 * A094386 A094387 A094388

KEYWORD

easy,nonn,tabl

AUTHOR

Philippe Deléham, Jun 03 2004, Jun 14 2007

STATUS

approved

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Last modified November 29 19:26 EST 2020. Contains 338769 sequences. (Running on oeis4.)