A121524 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 odd level (0 <= k <= n-1).

1, 1, 1, 1, 3, 1, 1, 6, 5, 1, 1, 9, 15, 8, 1, 1, 12, 34, 30, 11, 1, 1, 15, 62, 85, 55, 14, 1, 1, 18, 99, 200, 185, 89, 17, 1, 1, 21, 145, 402, 510, 365, 132, 20, 1, 1, 24, 200, 718, 1220, 1160, 650, 184, 23, 1, 1, 27, 264, 1175, 2585, 3155, 2400, 1067, 245, 26, 1, 1, 30, 337

Offset

1,5

Comments

A nondecreasing Dyck path is a Dyck path for which the sequence of the altitudes of the valleys is nondecreasing.

Row sums are the odd-indexed Fibonacci numbers (A001519).

T(n,k) = A121522(n,n-k), i.e., triangle is mirror image of A121522.

Sum_{k>=0} k*T(n,k) = A121525(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 q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.

Formula

G.f.: G(t,z) = z(1-tz^2)(1-2tz^2-t^2*z^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 UDU(U)D(U)DD, U(U)DDU(U)DD, U(U)D(U)UDDD, U(U)UDD(U)DD and U(U)U(U)DDDD, where U=(1,1) and D=(1,-1) (the up steps starting at an odd level are shown between parentheses; UUDUDDUD does not qualify because it is not nondecreasing).
Triangle starts:
  1;
  1,  1;
  1,  3,  1;
  1,  6,  5,  1;
  1,  9, 15,  8,  1;
  1, 12, 34, 30, 11,  1;

Maple

g:=z*(1-t*z^2)*(1-2*t*z^2-t^2*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=0..n-1) od; # yields sequence in triangular form

Mathematica

G[t_, z_] = z*(1 - t*z^2)*(1 - 2*t*z^2 - t^2*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);
T[n_, k_] := SeriesCoefficient[G[t, z], {z, 0, n}, {t, 0, k}];
Table[T[n, k], {n, 1, 12}, {k, 0, n - 1}] // Flatten (* Jean-François Alcover, Jan 15 2018 *)

Crossrefs

Cf. A001519, A121522, A121525.

Sequence in context: A178867 A335256 A102036 * A361682 A103141 A085478

Adjacent sequences: A121521 A121522 A121523 * A121525 A121526 A121527

Keyword

nonn,tabl

Author

Emeric Deutsch, Aug 05 2006

Status

approved