

A121484


Triangle read by rows: T(n,k) is the number of nondecreasing Dyck paths of semilength n, having k peaks at even level (n>=1,0<=k<=n1). A nondecreasing Dyck path is a Dyck path for which the sequence of the altitudes of the valleys is nondecreasing.


3



1, 1, 1, 2, 2, 1, 4, 6, 2, 1, 8, 13, 10, 2, 1, 16, 34, 23, 13, 2, 1, 33, 74, 75, 32, 16, 2, 1, 66, 178, 180, 124, 40, 19, 2, 1, 136, 390, 497, 321, 180, 48, 22, 2, 1, 274, 895, 1192, 1004, 488, 244, 56, 25, 2, 1, 562, 1958, 3033, 2598, 1701, 682, 317, 64, 28, 2, 1, 1138, 4374
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OFFSET

1,4


COMMENTS

Row sums are the oddsubscripted Fibonacci numbers (A001519). T(n,0)=A121485(n). Sum(k*T(n,k),k=0..n1)=A121486(n).


LINKS



FORMULA

G.f.: G(t,z) = z(1z^2)(12z^2tz^3)/(14z^2ztz+2tz^4+4z^4z^6 +2z^3+tz^3+t^2*z^3).


EXAMPLE

T(4,2)=2 because we have UDUUDUDD and UUDDUUDD, where U=(1,1) and D=(1,1) (the peaks at even level are shown by a ).
Triangle starts:
1;
1,1;
2,2,1;
4,6,2,1;
8,13,10,2,1;
16,34,23,13,2,1;


MAPLE

G:=z*(1z^2)*(12*z^2t*z^3)/(14*z^2zt*z+2*t*z^4+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..n1) od; # yields sequence in triangular form


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



