A109979 Triangle read by rows: T(n,k) (0<=k<=n) is the number of Delannoy paths of length n, having k (1,1)-steps on the line y=x (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)).

1, 2, 1, 8, 4, 1, 36, 20, 6, 1, 172, 104, 36, 8, 1, 852, 552, 212, 56, 10, 1, 4324, 2968, 1236, 368, 80, 12, 1, 22332, 16104, 7164, 2336, 580, 108, 14, 1, 116876, 87976, 41372, 14512, 3980, 856, 140, 16, 1, 618084, 483192, 238356, 88848, 26372, 6312, 1204, 176

Offset

0,2

Comments

Row sums are the central Delannoy numbers (A001850).

First column yields A109980.

sum(k*T(n,k),k=0..n) = A001109(n).

Links

Table of n, a(n) for n=0..52.

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

Formula

G.f.: [tz-z+sqrt(1-6z+z^2)]/(1-6z+2tz^2-t^2*z^2).

Example

T(2,1)=4 because we have DNE, DEN, NED and END.
Triangle begins:
1;
2,1;
8,4,1;
36,20,6,1;

Maple

G:=(t*z-z+sqrt(1-6*z+z^2))/(1-6*z+2*t*z^2-t^2*z^2): 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

Crossrefs

Cf. A001850, A109980, A001109.

Sequence in context: A154537 A201641 A110446 * A110171 A104988 A343296

Adjacent sequences: A109976 A109977 A109978 * A109980 A109981 A109982

Keyword

nonn,tabl

Author

Emeric Deutsch, Jul 06 2005

Status

approved