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A110171 Triangle read by rows: T(n,k) (0 <= k <= n) is the number of Delannoy paths of length n that start with exactly k (0,1) steps (or, equivalently, with exactly k (1,0) steps).

%I A110171 #37 Feb 17 2020 07:40:17
%S A110171 1,2,1,8,4,1,38,18,6,1,192,88,32,8,1,1002,450,170,50,10,1,5336,2364,
%T A110171 912,292,72,12,1,28814,12642,4942,1666,462,98,14,1,157184,68464,27008,
%U A110171 9424,2816,688,128,16,1,864146,374274,148626,53154,16722,4482,978,162,18,1
%N A110171 Triangle read by rows: T(n,k) (0 <= k <= n) is the number of Delannoy paths of length n that start with exactly k (0,1) steps (or, equivalently, with exactly k (1,0) steps).
%C A110171 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 A110171 Column k for k >= 1 has g.f. z^k*R^(k-1)*g*(1+z*R), where R = 1 + zR + zR^2 = (1 - z - sqrt(1-6z+z^2))/(2z) is the g.f. of the large Schroeder numbers (A006318) and g = 1/sqrt(1-6z+z^2) is the g.f. of the central Delannoy numbers (A001850).
%C A110171 Sum_{k=0..n} k*T(n,k) = A050151(n) (the partial sums of the central Delannoy numbers) = (1/2)*n*R(n), where R(n) = A006318(n) is the n-th large Schroeder number.
%C A110171 From _Paul Barry_, May 07 2009: (Start)
%C A110171 Riordan array ((1+x+sqrt(1-6x+x^2))/(2*sqrt(1-6x+x^2)), (1-x-sqrt(1-6x+x^2))/2).
%C A110171 Inverse of Riordan array ((1-2x-x^2)/(1-x^2), x(1-x)/(1+x)). (End)
%H A110171 G. C. Greubel, <a href="/A110171/b110171.txt">Rows n = 0..100 of triangle, flattened</a>
%H A110171 P. Peart and W.-J. Woan, <a href="http://dx.doi.org/10.1016/S0166-218X(99)00166-3">A divisibility property for a subgroup of Riordan matrices</a>, Discrete Applied Mathematics, Vol. 98, Issue 3, Jan 2000, 255-263.
%H A110171 Robert A. Sulanke, <a href="http://www.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.
%H A110171 W.-j. Woan, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Woan/woan655.html">The Lagrange Inversion Formula and Divisibility Properties</a>, JIS 10 (2007) 07.7.8, example 5.
%F A110171 T(n,0) = A002003(n) for n >= 1.
%F A110171 T(n,1) = A050146(n) for n >= 1.
%F A110171 Row sums are the central Delannoy numbers (A001850).
%F A110171 G.f.: (1+z+Q)/(Q(2-t+tz+tQ)), where Q=sqrt(1-6z+z^2).
%F A110171 T(n,k) = x^(n-k)*((1+x)/(1-x))^n. - _Paul Barry_, May 07 2009
%F A110171 T(n,k) = C(n, k)*hypergeometric([k-n, n], [k+1], -1). - _Peter Luschny_, Sep 17 2014
%F A110171 From _Peter Bala_, Jun 29 2015: (Start)
%F A110171 T(n,k) = Sum_{i = 0..n} binomial(n,i)*binomial(2*n-k-i-1,n-k-i).
%F A110171 Matrix product A118384 * A007318^(-1)
%F A110171 Riordan array has the form ( x*h'(x)/h(x), h(x) ) with h(x) = ( 1 - x - sqrt(1 - 6*x + x^2) )/2 and so belongs to the hitting time subgroup H of the Riordan group (see Peart and Woan). (End)
%F A110171 T(n,k) = P(n-k, k, -1, 3), where P(n, alpha, beta, x) is the n-th Jacobi polynomial with parameters alpha and beta. Cf. A113139. - _Peter Bala_, Feb 16 2020
%e A110171 T(2,1)=4 because we have NED, NENE, NEEN and NDE.
%e A110171 Triangle starts:
%e A110171     1;
%e A110171     2,  1;
%e A110171     8,  4,  1;
%e A110171    38, 18,  6,  1;
%e A110171   192, 88, 32,  8,  1;
%e A110171 From _Paul Barry_, May 07 2009: (Start)
%e A110171 Production matrix is
%e A110171    2, 1,
%e A110171    4, 2, 1,
%e A110171    6, 2, 2, 1,
%e A110171    8, 2, 2, 2, 1,
%e A110171   10, 2, 2, 2, 2, 1,
%e A110171   12, 2, 2, 2, 2, 2, 1,
%e A110171   14, 2, 2, 2, 2, 2, 2, 1,
%e A110171   16, 2, 2, 2, 2, 2, 2, 2, 1,
%e A110171   18, 2, 2, 2, 2, 2, 2, 2, 2, 1 (End)
%p A110171 Q:=sqrt(1-6*z+z^2): G:=(1+z+Q)/Q/(2-t+t*z+t*Q): 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
%t A110171 T[n_, n_] = 1;
%t A110171 T[n_, k_] := Sum[Binomial[n, i] Binomial[2n-k-i-1, n-k-i], {i, 0, n}];
%t A110171 Table[T[n, k], {n, 0, 9}, {k, 0, n}]//Flatten (* _Jean-François Alcover_, Jun 13 2019 *)
%o A110171 (Sage)
%o A110171 A110171 = lambda n,k : binomial(n, k)*hypergeometric([k-n, n], [k+1], -1)
%o A110171 for n in (0..9): [round(A110171(n,k).n(100)) for k in (0..n)] # _Peter Luschny_, Sep 17 2014
%Y A110171 Cf. A001850, A002003, A006318, A050146, A050151, A113139, A118384.
%K A110171 nonn,tabl
%O A110171 0,2
%A A110171 _Emeric Deutsch_, Jul 14 2005