|
%I #20 Feb 24 2020 00:38:44
%S 1,3,12,1,53,10,247,73,1,1192,474,17,5897,2908,183,1,29723,17290,1602,
%T 24,152020,100891,12475,342,1,786733,581814,90205,3780,31,4111295,
%U 3329507,620243,35857,550,1,21661168,18956564,4114406,307192,7351,38
%N Triangle read by rows: T(n,k) (0 <= k <= floor(n/2)) is the number of Delannoy paths of length n, having k EE's crossing the line y = x (i.e., two consecutive E steps from the line y = x+1 to the line y = x-1).
%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 contains 1 + floor(n/2) terms.
%C Row sums are the central Delannoy numbers (A001850).
%H Jinyuan Wang, <a href="/A110121/b110121.txt">Rows n = 0..50 of triangle, flattened</a>
%H Andrei Asinowski, Axel Bacher, Cyril Banderier, Bernhard Gittenberger, <a href="https://lipn.univ-paris13.fr/~banderier/Papers/patterns2019.pdf">Analytic combinatorics of lattice paths with forbidden patterns, the vectorial kernel method, and generating functions for pushdown automata</a>, Laboratoire d'Informatique de Paris Nord (LIPN 2019).
%H R. A. Sulanke, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Sulanke/delannoy.html">Objects counted by the central Delannoy numbers</a>, J. Integer Seq. 6 (2003), no. 1, Article 03.1.5.
%F T(n,0) = A110122(n).
%F Sum_{k=0..floor(n/2)} k*T(n,k) = A110127(n).
%F G.f.: 1/((1 - zR)^2 - z - tz^2*R^2), where R = 1 + zR + zR^2 = (1 - z - sqrt(1 - 6z + z^2))/(2z) is the g.f. of the large Schroeder numbers (A006318).
%e T(2,0)=12 because, among the 13 (=A001850(2)) Delannoy paths of length 2, only NEEN has an EE crossing the line y=x.
%e Triangle begins:
%e 1;
%e 3;
%e 12, 1;
%e 53, 10;
%e 247, 73, 1;
%p R:=(1-z-sqrt(1-6*z+z^2))/2/z: G:=1/((1-z*R)^2-z-t*z^2*R^2): 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
%t nmax = 11; r := (1 - z - Sqrt[1 - 6*z + z^2])/2/z; g := 1/((1 - z*r)^2 - z - t*z^2*r^2); gser = Series[g, {z, 0, nmax}]; p[0] = 1; Do[ p[n] = Coefficient[ gser, z, n] , {n, 1, nmax}]; row[n_] := Table[ Coefficient[ t*p[n], t, k], {k, 1, 1 + Floor[n/2]}]; Flatten[ Table[ row[n], {n, 0, nmax}]] (* _Jean-François Alcover_, Dec 07 2011, after Maple *)
%Y Cf. A001850, A006318, A110122, A110123, A110127.
%K nonn,tabf
%O 0,2
%A _Emeric Deutsch_, Jul 13 2005
|
