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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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,5
COMMENTS
Row sums are the odd-subscripted Fibonacci numbers (A001519). T(n,k)=A121524(n,n-k), i.e. triangle is mirror image of A121524. Sum(k*T(n,k), k=1..n)=A121523(n).
LINKS
E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.
FORMULA
G.f.=G(t,z)=tz(1-tz^2)(1-2tz^2-tz^3)/(1-z-tz-4tz^2+2tz^3+2t^2*z^3+6t^2*z^4-t^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*(1-t*z^2)*(1-2*t*z^2-t*z^3)/(1-z-t*z-4*t*z^2+2*t*z^3+2*t^2*z^3+6*t^2*z^4-t^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
Sequence in context: A076756 A114172 A271942 * A294582 A294589 A204027
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Aug 05 2006
STATUS
approved

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Last modified May 23 20:34 EDT 2024. Contains 372765 sequences. (Running on oeis4.)