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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<=n-1). 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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Row sums are the odd-subscripted Fibonacci numbers (A001519). T(n,0)=A121485(n). Sum(k*T(n,k),k=0..n-1)=A121486(n).

LINKS

Table of n, a(n) for n=1..68.

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-z^2)(1-2z^2-tz^3)/(1-4z^2-z-tz+2tz^4+4z^4-z^6 +2z^3+tz^3+t^2*z^3).

EXAMPLE

T(4,2)=2 because we have UDUU|DU|DD and UU|DDUU|DD, 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*(1-z^2)*(1-2*z^2-t*z^3)/(1-4*z^2-z-t*z+2*t*z^4+4*z^4-z^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-1) od; # yields sequence in triangular form

CROSSREFS

Cf. A001519, A121481, A121485, A121486.

Sequence in context: A330843 A115313 A048942 * A273903 A346420 A080928

Adjacent sequences: A121481 A121482 A121483 * A121485 A121486 A121487

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Aug 02 2006

STATUS

approved

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Last modified January 31 18:34 EST 2023. Contains 359980 sequences. (Running on oeis4.)