

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



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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)



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.: [tzz+sqrt(16z+z^2)]/(16z+2tz^2t^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*zz+sqrt(16*z+z^2))/(16*z+2*t*z^2t^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 A136225
Adjacent sequences: A109976 A109977 A109978 * A109980 A109981 A109982


KEYWORD

nonn,tabl


AUTHOR

Emeric Deutsch, Jul 06 2005


STATUS

approved



