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

%I A109983 #30 Sep 03 2024 21:01:56
%S A109983 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 A109983 1,30,210,560,630,252,0,0,0,0,0,0,1,42,420,1680,3150,2772,924,0,0,0,0,
%U A109983 0,0,0,1,56,756,4200,11550,16632,12012,3432
%N A109983 Triangle read by rows: T(n, k) (0<=k<=2n) is the number of Delannoy paths of length n, having k steps.
%C A109983 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 A109983 Row n has 2*n+1 terms, the first n of which are 0.
%C A109983 Row sums are the central Delannoy numbers (A001850).
%C A109983 Column sums are the central trinomial coefficients (A002426).
%H A109983 Reinhard Zumkeller, <a href="/A109983/b109983.txt">Rows n = 0..100 of triangle, flattened</a>
%H A109983 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 A109983 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 A109983 T(n, k) = binomial(n, 2*n-k) binomial(k, n).
%F A109983 T(n, k) = A104684(n, 2*n-k).
%F A109983 G.f.: 1/sqrt((1 - t*z)^2 - 4*z*t^2).
%F A109983 T(n, 2*n) = binomial(2*n, n) (A000984).
%F A109983 Sum_{k=0..n} k*T(n, k) = A109984(n).
%F A109983 T(n, k) = A063007(n, k-n). - _Michael Somos_, Sep 22 2013
%e A109983 T(2, 3) = 6 because we have DNE, DEN, NED, END, NDE and EDN.
%e A109983 Triangle begins
%e A109983   1;
%e A109983   0,1,2;
%e A109983   0,0,1,6,6;
%e A109983   0,0,0,1,12,30,20;
%e A109983   ...
%p A109983 T := (n,k)->binomial(n,2*n-k)*binomial(k,n):
%p A109983 for n from 0 to 8 do seq(T(n,k),k=0..2*n) od; # yields sequence in triangular form
%p A109983 # Alternative:
%p A109983 gf := ((1 - x*y)^2 - 4*x^2*y)^(-1/2):
%p A109983 yser := series(gf, y, 12): ycoeff := n -> coeff(yser, y, n):
%p A109983 row := n -> seq(coeff(expand(ycoeff(n)), x, k), k=0..2*n):
%p A109983 seq(row(n), n=0..7); # _Peter Luschny_, Oct 28 2020
%o A109983 (PARI) {T(n, k) = binomial(n, k-n) * binomial(k, n)} /* _Michael Somos_, Sep 22 2013 */
%o A109983 (Haskell)
%o A109983 a109983 n k = a109983_tabf !! n !! k
%o A109983 a109983_row n = a109983_tabf !! n
%o A109983 a109983_tabf = zipWith (++) (map (flip take (repeat 0)) [0..]) a063007_tabl
%o A109983 -- _Reinhard Zumkeller_, Nov 18 2014
%Y A109983 Cf. A001850, A002426, A000984, A063007, A104684, A109984.
%K A109983 nonn,tabf
%O A109983 0,4
%A A109983 _Emeric Deutsch_, Jul 07 2005