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A110221 Triangle read by rows: T(n,k) (0<=k<=floor(n/2)) is the number of Delannoy paths of length n, having k ED's. 0
1, 3, 11, 2, 45, 18, 195, 120, 6, 873, 720, 90, 3989, 4110, 870, 20, 18483, 22806, 6930, 420, 86515, 124264, 49560, 5320, 70, 408105, 668520, 331128, 52920, 1890, 1936881, 3562830, 2111760, 456120, 29610, 252, 9238023, 18850590, 13020480, 3575880 (list; graph; refs; listen; history; text; internal format)
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).

Row n has 1+floor(n/2) terms. Row sums are the central Delannoy numbers (A001850). Column 0 yields A026375. Sum(k*T(n,k),k=0..floor(n/2))=2*A002695(n).

LINKS

Table of n, a(n) for n=0..39.

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

FORMULA

G.f.: 1/(1-z-2tz^2*R-2zR+2z^2*R), where R=[1-z-sqrt(1-6z+5z^2-4tz^2)]/[2z(1-z+tz)].

EXAMPLE

T(2,1)=2 because we have NED and EDN.

Triangle begins:

1;

3;

11,2;

45,18;

195,120,6;

MAPLE

R:=(1-z-sqrt(1-6*z+5*z^2-4*z^2*t))/2/z/(1-z+t*z): G:=1/(1-z-2*t*z^2*R-2*z*R+2*z^2*R): Gser:=simplify(series(G, z=0, 15)): P[0]:=1: for n from 1 to 12 do P[n]:=coeff(Gser, z^n) od: for n from 0 to 12 do seq(coeff(t*P[n], t^k), k=1..1+floor(n/2)) od; # yields sequence in triangular form

CROSSREFS

Cf. A001850, A026375, A002695.

Sequence in context: A302120 A133369 A110123 * A244237 A238683 A303114

Adjacent sequences:  A110218 A110219 A110220 * A110222 A110223 A110224

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Jul 20 2005

STATUS

approved

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Last modified May 12 02:06 EDT 2021. Contains 343808 sequences. (Running on oeis4.)