A110171 Triangle read by rows: T(n,k) (0 <= k <= n) is the number of Delannoy paths of length n that start with exactly k (0,1) steps (or, equivalently, with exactly k (1,0) steps).

1, 2, 1, 8, 4, 1, 38, 18, 6, 1, 192, 88, 32, 8, 1, 1002, 450, 170, 50, 10, 1, 5336, 2364, 912, 292, 72, 12, 1, 28814, 12642, 4942, 1666, 462, 98, 14, 1, 157184, 68464, 27008, 9424, 2816, 688, 128, 16, 1, 864146, 374274, 148626, 53154, 16722, 4482, 978, 162, 18, 1

Offset

0,2

Comments

A Delannoy path of length n is a path from (0,0) to (n,n), consisting of steps E=(1,0), N=(0,1) and D=(1,1).

Column k for k >= 1 has g.f. z^k*R^(k-1)*g*(1+z*R), where R = 1 + zR + zR^2 = (1 - z - sqrt(1-6z+z^2))/(2z) is the g.f. of the large Schroeder numbers (A006318) and g = 1/sqrt(1-6z+z^2) is the g.f. of the central Delannoy numbers (A001850).

Sum_{k=0..n} k*T(n,k) = A050151(n) (the partial sums of the central Delannoy numbers) = (1/2)*n*R(n), where R(n) = A006318(n) is the n-th large Schroeder number.

From Paul Barry, May 07 2009: (Start)

Riordan array ((1+x+sqrt(1-6x+x^2))/(2*sqrt(1-6x+x^2)), (1-x-sqrt(1-6x+x^2))/2).

Inverse of Riordan array ((1-2x-x^2)/(1-x^2), x(1-x)/(1+x)). (End)

Links

G. C. Greubel, Rows n = 0..100 of triangle, flattened

P. Peart and W.-J. Woan, A divisibility property for a subgroup of Riordan matrices, Discrete Applied Mathematics, Vol. 98, Issue 3, Jan 2000, 255-263.

Robert A. Sulanke, Objects Counted by the Central Delannoy Numbers, Journal of Integer Sequences, Volume 6, 2003, Article 03.1.5.

W.-j. Woan, The Lagrange Inversion Formula and Divisibility Properties, JIS 10 (2007) 07.7.8, example 5.

Formula

T(n,0) = A002003(n) for n >= 1.

T(n,1) = A050146(n) for n >= 1.

Row sums are the central Delannoy numbers (A001850).

G.f.: (1+z+Q)/(Q(2-t+tz+tQ)), where Q=sqrt(1-6z+z^2).

T(n,k) = x^(n-k)*((1+x)/(1-x))^n. - Paul Barry, May 07 2009

T(n,k) = C(n, k)*hypergeometric([k-n, n], [k+1], -1). - Peter Luschny, Sep 17 2014

From Peter Bala, Jun 29 2015: (Start)

T(n,k) = Sum_{i = 0..n} binomial(n,i)*binomial(2*n-k-i-1,n-k-i).

Matrix product A118384 * A007318^(-1)

Riordan array has the form ( x*h'(x)/h(x), h(x) ) with h(x) = ( 1 - x - sqrt(1 - 6*x + x^2) )/2 and so belongs to the hitting time subgroup H of the Riordan group (see Peart and Woan). (End)

T(n,k) = P(n-k, k, -1, 3), where P(n, alpha, beta, x) is the n-th Jacobi polynomial with parameters alpha and beta. Cf. A113139. - Peter Bala, Feb 16 2020

Example

T(2,1)=4 because we have NED, NENE, NEEN and NDE.
Triangle starts:
    1;
    2,  1;
    8,  4,  1;
   38, 18,  6,  1;
  192, 88, 32,  8,  1;
From Paul Barry, May 07 2009: (Start)
Production matrix is
   2, 1,
   4, 2, 1,
   6, 2, 2, 1,
   8, 2, 2, 2, 1,
  10, 2, 2, 2, 2, 1,
  12, 2, 2, 2, 2, 2, 1,
  14, 2, 2, 2, 2, 2, 2, 1,
  16, 2, 2, 2, 2, 2, 2, 2, 1,
  18, 2, 2, 2, 2, 2, 2, 2, 2, 1 (End)

Maple

Q:=sqrt(1-6*z+z^2): G:=(1+z+Q)/Q/(2-t+t*z+t*Q): Gser:=simplify(series(G, z=0, 13)): P[0]:=1: for n from 1 to 10 do P[n]:=coeff(Gser, z^n) od: for n from 0 to 10 do seq(coeff(t*P[n], t^k), k=1..n+1) od; # yields sequence in triangular form

Mathematica

T[n_, n_] = 1;
T[n_, k_] := Sum[Binomial[n, i] Binomial[2n-k-i-1, n-k-i], {i, 0, n}];
Table[T[n, k], {n, 0, 9}, {k, 0, n}]//Flatten (* Jean-François Alcover, Jun 13 2019 *)

Prog

(Sage)
A110171 = lambda n, k : binomial(n, k)*hypergeometric([k-n, n], [k+1], -1)
for n in (0..9): [round(A110171(n, k).n(100)) for k in (0..n)] # Peter Luschny, Sep 17 2014

Crossrefs

Cf. A001850, A002003, A006318, A050146, A050151, A113139, A118384.

Sequence in context: A201641 A110446 A109979 * A104988 A343296 A136225

Adjacent sequences: A110168 A110169 A110170 * A110172 A110173 A110174

Keyword

nonn,tabl

Author

Emeric Deutsch, Jul 14 2005

Status

approved