

A121522


Triangle read by rows: T(n,k) is the number of nondecreasing Dyck paths of semilength n and having k up steps starting at an even level (1<=k<=n). A nondecreasing Dyck path is a Dyck path for which the sequence of the altitudes of the valleys is nondecreasing.


2



1, 1, 1, 1, 3, 1, 1, 5, 6, 1, 1, 8, 15, 9, 1, 1, 11, 30, 34, 12, 1, 1, 14, 55, 85, 62, 15, 1, 1, 17, 89, 185, 200, 99, 18, 1, 1, 20, 132, 365, 510, 402, 145, 21, 1, 1, 23, 184, 650, 1160, 1220, 718, 200, 24, 1, 1, 26, 245, 1067, 2400, 3155, 2585, 1175, 264, 27, 1, 1, 29, 315
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OFFSET

1,5


COMMENTS

Row sums are the oddsubscripted Fibonacci numbers (A001519). T(n,k)=A121524(n,nk), i.e. triangle is mirror image of A121524. Sum(k*T(n,k), k=1..n)=A121523(n).


LINKS

Table of n, a(n) for n=1..69.
E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, Nondecreasing Dyck paths and qFibonacci numbers, Discrete Math., 170, 1997, 211217.


FORMULA

G.f.=G(t,z)=tz(1tz^2)(12tz^2tz^3)/(1ztz4tz^2+2tz^3+2t^2*z^3+6t^2*z^4t^3*z^6).


EXAMPLE

T(4,2)=5 because we have (U)D(U)UDUDD, (U)UDD(U)UDD, (U)UDU(U)DDD, (U)U(U)DDUDD and (U)U(U)UDDDD, where U=(1,1) and D=(1,1) (the up steps starting at an even level are shown between parentheses; UUDUDDUD does not qualify because it is not nondecreasing).
Triangle starts:
1;
1,1;
1,3,1;
1,5,6,1;
1,8,15,9,1;
1,11,30,34,12,1;


MAPLE

g:=t*z*(1t*z^2)*(12*t*z^2t*z^3)/(1zt*z4*t*z^2+2*t*z^3+2*t^2*z^3+6*t^2*z^4t^3*z^6): gser:=simplify(series(g, z=0, 17)): for n from 1 to 12 do P[n]:=sort(expand(coeff(gser, z, n))) od: for n from 1 to 12 do seq(coeff(P[n], t, j), j=1..n) od; # yields sequence in triangular form


CROSSREFS

Cf. A001519, A121523, A121524.
Sequence in context: A076756 A114172 A271942 * A294582 A294589 A204027
Adjacent sequences: A121519 A121520 A121521 * A121523 A121524 A121525


KEYWORD

nonn,tabl


AUTHOR

Emeric Deutsch, Aug 05 2006


STATUS

approved



