OFFSET
0,6
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.
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)
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 *)
PROG
(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
(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
CROSSREFS
KEYWORD
AUTHOR
Philippe Deléham, Jun 03 2004, Jun 14 2007
EXTENSIONS
New name using a formula of the author by Peter Luschny, Sep 26 2024
STATUS
approved