

A110189


Triangle read by rows: T(n,k) (0<=k<=n) is the number of Schroeder paths of length 2n, having k (1,0)steps on the lines y=0 and y=1 (a Schroeder path of length 2n is a path from (0,0) to (2n,0), consisting of steps U=(1,1), D=(1,1) and H=(2,0) and never going below the xaxis).


1



1, 1, 1, 2, 3, 1, 6, 9, 6, 1, 22, 32, 25, 10, 1, 90, 128, 105, 55, 15, 1, 394, 552, 462, 271, 105, 21, 1, 1806, 2504, 2118, 1317, 602, 182, 28, 1, 8558, 11776, 10026, 6456, 3235, 1204, 294, 36, 1, 41586, 56896, 48658, 32068, 17019, 7149, 2226, 450, 45, 1, 206098
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OFFSET

0,4


COMMENTS

Row sums are the large Schroeder numbers (A006318). First column yields the large Schroeder numbers (A006318). sum(k*T(n,k),k=0..n)=A110190(n).


LINKS

Table of n, a(n) for n=0..55.


FORMULA

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


EXAMPLE

T(3,2)=6 because we have HHUD, HUHD, HUDH, UDHH, UHDH and UHHD.
Triangle starts:
1;
1,1;
2,3,1;
6,9,6,1;
22,32,25,10,1;


MAPLE

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


CROSSREFS

Cf. A006318, A110190.
Sequence in context: A198427 A289905 A086211 * A187914 A321625 A132372
Adjacent sequences: A110186 A110187 A110188 * A110190 A110191 A110192


KEYWORD

nonn,tabl


AUTHOR

Emeric Deutsch, Jul 15 2005


STATUS

approved



