A109983 Triangle read by rows: T(n, k) (0<=k<=2n) is the number of Delannoy paths of length n, having k steps.

1, 0, 1, 2, 0, 0, 1, 6, 6, 0, 0, 0, 1, 12, 30, 20, 0, 0, 0, 0, 1, 20, 90, 140, 70, 0, 0, 0, 0, 0, 1, 30, 210, 560, 630, 252, 0, 0, 0, 0, 0, 0, 1, 42, 420, 1680, 3150, 2772, 924, 0, 0, 0, 0, 0, 0, 0, 1, 56, 756, 4200, 11550, 16632, 12012, 3432

Offset

0,4

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 2*n+1 terms, the first n of which are 0.

Row sums are the central Delannoy numbers (A001850).

Column sums are the central trinomial coefficients (A002426).

Links

Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened

Hsien-Kuei Hwang and Satoshi Kuriki, Integrated empirical measures and generalizations of classical goodness-of-fit statistics, arXiv:2404.06040 [math.ST], 2024. See p. 11.

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

Formula

T(n, k) = binomial(n, 2*n-k) binomial(k, n).

T(n, k) = A104684(n, 2*n-k).

G.f.: 1/sqrt((1 - t*z)^2 - 4*z*t^2).

T(n, 2*n) = binomial(2*n, n) (A000984).

Sum_{k=0..n} k*T(n, k) = A109984(n).

T(n, k) = A063007(n, k-n). - Michael Somos, Sep 22 2013

Example

T(2, 3) = 6 because we have DNE, DEN, NED, END, NDE and EDN.
Triangle begins
  1;
  0,1,2;
  0,0,1,6,6;
  0,0,0,1,12,30,20;
  ...

Maple

T := (n, k)->binomial(n, 2*n-k)*binomial(k, n):
for n from 0 to 8 do seq(T(n, k), k=0..2*n) od; # yields sequence in triangular form
# Alternative:
gf := ((1 - x*y)^2 - 4*x^2*y)^(-1/2):
yser := series(gf, y, 12): ycoeff := n -> coeff(yser, y, n):
row := n -> seq(coeff(expand(ycoeff(n)), x, k), k=0..2*n):
seq(row(n), n=0..7); # Peter Luschny, Oct 28 2020

Prog

(PARI) {T(n, k) = binomial(n, k-n) * binomial(k, n)} /* Michael Somos, Sep 22 2013 */
(Haskell)
a109983 n k = a109983_tabf !! n !! k
a109983_row n = a109983_tabf !! n
a109983_tabf = zipWith (++) (map (flip take (repeat 0)) [0..]) a063007_tabl
-- Reinhard Zumkeller, Nov 18 2014

Crossrefs

Cf. A001850, A002426, A000984, A063007, A104684, A109984.

Sequence in context: A113129 A127826 A228866 * A332409 A367000 A289084

Adjacent sequences: A109980 A109981 A109982 * A109984 A109985 A109986

Keyword

nonn,tabf,changed

Author

Emeric Deutsch, Jul 07 2005

Status

approved