A094385 Triangle read by rows: T(n, k) = binomial(2*n, k-1)*binomial(2*n-k-1, n-k)/n for n, k >= 1, and T(n, 0) = 0^n.

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

Author

Philippe Deléham, Jun 03 2004, Jun 14 2007

Keywords

• easy
• nonn
• tabl

Examples

			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; ...
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened
• 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.

Crossrefs

Variant of A062991, unsigned and transposed.

See also A234950 for another version.

Columns: A000007 (k=0), 2*A001700 (k=1).

Diagonals: A002694 (k=n-1), A000108 (k=n).

Row sums: A064062 (generalized Catalan C(2; n)).

Cf. A000012, A002694, A064062, A064063, A064087, A064088, A064089.

Cf. A064090, A064091, A064092, A064093, A064094.

Programs

Magma

A094385:= func< n,k | n eq 0 select 1 else Binomial(2*n, k-1)*Binomial(2*n-k-1, n-k)/n >;
[A094385(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 26 2024

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 *)

SageMath

def A094385(n,k): return 1 if (n==0) else binomial(2*n,k-1)*binomial(2*n-k-1, n-k)//n
flatten([[A094385(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 26 2024

Formula

T is 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.

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, A064063, A064087..A064093 for x = -1, 0, ..., 9, respectively.

From G. C. Greubel, Sep 26 2024: (Start)

T(n, 1) = A000108(n-1), n >= 1.

T(n, n-1) = A002694(n), n >= 1.

T(n, n) = A000108(n). (End)

Extensions

New name using a formula of the author by Peter Luschny, Sep 26 2024