

A121481


Triangle read by rows: T(n,k) is the number of nondecreasing Dyck paths of semilength n, having k peaks at odd level (0<=k<=n).


3



1, 0, 1, 1, 0, 1, 1, 3, 0, 1, 3, 3, 6, 0, 1, 5, 14, 5, 9, 0, 1, 12, 22, 35, 7, 12, 0, 1, 22, 68, 53, 65, 9, 15, 0, 1, 49, 127, 203, 97, 104, 11, 18, 0, 1, 94, 329, 390, 444, 153, 152, 13, 21, 0, 1, 201, 664, 1157, 873, 816, 221, 209, 15, 24, 0, 1, 396, 1576, 2456, 2925, 1627, 1345
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OFFSET

0,8


COMMENTS

Row sums are the oddsubscripted Fibonacci numbers (A001519). T(n,0)=A121482(n). Sum(k*T(n,k),k=0..n)=A121483(n).


LINKS



FORMULA

G.f.: G(t,z) = (tz^2+z^2+z1)(tz^3+2z^21)/(14z^2ztz+2tz^4+4z^4z^6+2z^3+tz^3+t^2*z^3).


EXAMPLE

T(3,1)=3 because we have UDUUDUDD, UUDUUDDD and UUUDDUDD, where U=(1,1) and D=(1,1) (the peaks at odd level are shown by a ; the Dyck path UUDUDDUD has 1 peak at odd level but it is not nondecreasing).
Triangle starts:
1;
0,1;
1,0,1;
1,3,0,1;
3,3,6,0,1;
5,14,5,9,0,1;


MAPLE

G:=(t*z^2+z^2+z1)*(t*z^3+2*z^21)/(14*z^2zt*z+2*z^4*t+4*z^4z^6+2*z^3+t*z^3+z^3*t^2): Gser:=simplify(series(G, z=0, 15)): P[0]:=1: for n from 1 to 12 do P[n]:=sort(coeff(Gser, z^n)) od: for n from 0 to 12 do seq(coeff(P[n], t, j), j=0..n) od; # yields sequence in triangular form


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



