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Triangle read by rows: T(n, k) (0<=k<=2n) is the number of Delannoy paths of length n, having k steps.
6

%I #30 Sep 03 2024 21:01:56

%S 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,

%T 1,30,210,560,630,252,0,0,0,0,0,0,1,42,420,1680,3150,2772,924,0,0,0,0,

%U 0,0,0,1,56,756,4200,11550,16632,12012,3432

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

%C 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).

%C Row n has 2*n+1 terms, the first n of which are 0.

%C Row sums are the central Delannoy numbers (A001850).

%C Column sums are the central trinomial coefficients (A002426).

%H Reinhard Zumkeller, <a href="/A109983/b109983.txt">Rows n = 0..100 of triangle, flattened</a>

%H Hsien-Kuei Hwang and Satoshi Kuriki, <a href="https://arxiv.org/abs/2404.06040">Integrated empirical measures and generalizations of classical goodness-of-fit statistics</a>, arXiv:2404.06040 [math.ST], 2024. See p. 11.

%H Robert A. Sulanke, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL6/Sulanke/delannoy.html">Objects Counted by the Central Delannoy Numbers</a>, Journal of Integer Sequences, Volume 6, 2003, Article 03.1.5.

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

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

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

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

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

%F T(n, k) = A063007(n, k-n). - _Michael Somos_, Sep 22 2013

%e T(2, 3) = 6 because we have DNE, DEN, NED, END, NDE and EDN.

%e Triangle begins

%e 1;

%e 0,1,2;

%e 0,0,1,6,6;

%e 0,0,0,1,12,30,20;

%e ...

%p T := (n,k)->binomial(n,2*n-k)*binomial(k,n):

%p for n from 0 to 8 do seq(T(n,k),k=0..2*n) od; # yields sequence in triangular form

%p # Alternative:

%p gf := ((1 - x*y)^2 - 4*x^2*y)^(-1/2):

%p yser := series(gf, y, 12): ycoeff := n -> coeff(yser, y, n):

%p row := n -> seq(coeff(expand(ycoeff(n)), x, k), k=0..2*n):

%p seq(row(n), n=0..7); # _Peter Luschny_, Oct 28 2020

%o (PARI) {T(n, k) = binomial(n, k-n) * binomial(k, n)} /* _Michael Somos_, Sep 22 2013 */

%o (Haskell)

%o a109983 n k = a109983_tabf !! n !! k

%o a109983_row n = a109983_tabf !! n

%o a109983_tabf = zipWith (++) (map (flip take (repeat 0)) [0..]) a063007_tabl

%o -- _Reinhard Zumkeller_, Nov 18 2014

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

%K nonn,tabf

%O 0,4

%A _Emeric Deutsch_, Jul 07 2005