A064094 Triangle composed of generalized Catalan numbers.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 3, 1, 1, 1, 14, 13, 4, 1, 1, 1, 42, 67, 25, 5, 1, 1, 1, 132, 381, 190, 41, 6, 1, 1, 1, 429, 2307, 1606, 413, 61, 7, 1, 1, 1, 1430, 14589, 14506, 4641, 766, 85, 8, 1, 1, 1, 4862, 95235, 137089, 55797, 10746, 1279, 113, 9, 1, 1

Offset

0,8

Comments

The column m sequence (without leading zeros and the first 1) appears in the Derrida et al. 1992 reference as Z_{N}=Y_{N}(N+1), N >=0, for alpha = m, beta = 1 (or alpha = 1, beta = m). In the Derrida et al. 1993 reference the formula in eq. (39) gives Z_{N}(alpha,beta)/(alpha*beta)^N for N>=1.

Links

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

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

B. Derrida, M. R. Evans, V. Hakim, and V. Pasquier, Exact solution of a 1D asymmetric exclusion model using a matrix formulation, J. Phys. A 26, 1993, 1493-1517; eq. (39), p. 1501, also appendix A1, (A12) p. 1513.

Formula

G.f. for column m: (x^m)/(1-x*c(m*x))= (x^m)*((m-1)+m*x*c(m*x))/(m-1+x) with the g.f. c(x) of Catalan numbers A000108.

T(n, m) = Sum_{j=0..n-m-1} ((n-m-j)*binomial(n-m-1+j, j)*(m^j)/(n-m) or T(n, m) = (1/(1-m))^(n-m)*(1 - m*Sum_{j=0..n-m-1} C(j)*(m*(1-m))^j ), for n - m >= 1, T(n, n) = 1, T(n, m) = 0 if n<m; with C(k) = A000108(k) (Catalan).

Example

Triangle begins:
  1;
  1,    1;
  1,    1,     1;
  1,    2,     1,     1;
  1,    5,     3,     1,    1;
  1,   14,    13,     4,    1,   1;
  1,   42,    67,    25,    5,   1,   1;
  1,  132,   381,   190,   41,   6,   1,   1;
  1,  429,  2307,  1606,  413,  61,   7,   1,   1;
  1, 1430, 14589, 14506, 4641, 766,  85,   8,   1,   1;

Mathematica

T[n_, 0] = 1; T[n_, 1] := CatalanNumber[n - 1]; T[n_, n_] = 1; T[n_, m_] := (1/(1 - m))^(n - m)*(1 - m*Sum[ CatalanNumber[k]*(m*(1 - m))^k, {k, 0, n - m - 1}]); Table[ T[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jul 05 2013 *)

Prog

(Magma)
function A064094(n, k)
  if k eq 0 or k eq n then return 1;
  else return (&+[(n-k-j)*Binomial(n-k-1+j, j)*k^j: j in [0..n-k-1]])/(n-k);
  end if;
end function;
[A064094(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 27 2024
(SageMath)
def A064094(n, k):
    if (k==0 or k==n): return 1
    else: return sum((n-k-j)*binomial(n-k-1+j, j)*k^j for j in range(n-k))//(n-k)
flatten([[A064094(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 27 2024

Crossrefs

Diagonals: A000012, A000012, A000027, A001844, A064096, A064302, A064303, A064304, A064305.

Columns (without leading zeros): A000012 (k=0), A000108 (k=1), A064062 (k=2), A064063 (k=3), A064087 (k=4), A064088 (k=5), A064089 (k=6), A064090 (k=7), A064091 (k=8), A064092 (k=9), A064093 (k=10).

Cf. A064095 (row sums).

Sequence in context: A069739 A066060 A008550 * A090182 A256384 A111673

Adjacent sequences: A064091 A064092 A064093 * A064095 A064096 A064097

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang, Sep 13 2001

Status

approved