

A110121


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=x1; 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)).


3



1, 3, 12, 1, 53, 10, 247, 73, 1, 1192, 474, 17, 5897, 2908, 183, 1, 29723, 17290, 1602, 24, 152020, 100891, 12475, 342, 1, 786733, 581814, 90205, 3780, 31, 4111295, 3329507, 620243, 35857, 550, 1, 21661168, 18956564, 4114406, 307192, 7351, 38
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,2


COMMENTS

Row n contains 1+floor(n/2) terms. Row sums are the central Delannoy numbers (A001850). T(n,0)=A110122(n). sum(k*T(n,k),k=0..floor(n/2))=A110127(n).


REFERENCES

R. A. Sulanke, Objects counted by the central Delannoy numbers, J. of Integer Sequences, 6, 2003, Article 03.1.5.


LINKS

Table of n, a(n) for n=0..41.


FORMULA

G.f.=1/[(1zR)^2ztz^2*R^2], where R=1+zR+zR^2=[1zsqrt(16z+z^2)]/(2z) is the g.f. of the large Schroeder numbers (A006318).


EXAMPLE

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.
Triangle begins:
1;
3;
12,1;
53,10;
247,73,1;


MAPLE

R:=(1zsqrt(16*z+z^2))/2/z: G:=1/((1z*R)^2zt*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


MATHEMATICA

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}]] (* JeanFrançois Alcover, Dec 07 2011, after Maple *)


CROSSREFS

Cf. A006318, A001850, A110122, A110123, A110127.
Sequence in context: A072117 A162853 A162854 * A069522 A170857 A227106
Adjacent sequences: A110118 A110119 A110120 * A110122 A110123 A110124


KEYWORD

nonn,tabf


AUTHOR

Emeric Deutsch, Jul 13 2005


STATUS

approved



