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.

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

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.

Paul 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 and Catherine Yan, Counting with Borel's triangle, Texas A&M University.

Yue Cai and 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.

Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021.

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: A185197 A327116 A157491 * A355260 A291799 A295027

Adjacent sequences: A094382 A094383 A094384 * A094386 A094387 A094388

Keyword

easy,nonn,tabl

Author

Philippe Deléham, Jun 03 2004, Jun 14 2007

Status

approved