0,8

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.

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.

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).

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;

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 *)

(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

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

nonn,easy,tabl

Wolfdieter Lang, Sep 13 2001

approved