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A121529 Triangle read by rows: T(n,k) is the number of nondecreasing Dyck paths of semilength n and having k double rises at an odd level (n>=1, k>=0). 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, 4, 1, 10, 2, 1, 19, 14, 1, 33, 50, 5, 1, 55, 132, 45, 1, 90, 301, 205, 13, 1, 146, 631, 680, 139, 1, 236, 1255, 1892, 763, 34, 1, 381, 2409, 4717, 3019, 419, 1, 615, 4509, 10920, 9846, 2677, 89, 1, 993, 8283, 23974, 28292, 12241, 1241, 1, 1604, 14998 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Row n contains 1+floor(n/2) terms. Row sums are the odd-subscripted Fibonacci numbers (A001519). T(2n,n)=Fibonacci(2n-1) (A001519). Sum(k*T(n,k), k>=0)=A121530(n).

REFERENCES

E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.

LINKS

Table of n, a(n) for n=1..58.

FORMULA

G.f.=G(t,z)=z(1-tz^2)(1-z+tz-z^2-tz^2-t^2*z^3)/[(1-z-tz^2)(1-z-z^2-3tz^2-tz^3+t^2*z^4)].

EXAMPLE

T(4,2)=2 because we have U/UDDU/UDD and U/UU/UDDDD, where U=(1,1) and D=(1,-1) (the double rises at an odd level are indicated by a /).

Triangle starts:

1;

1,1;

1,4;

1,10,2;

1,19,14;

1,33,50,5;

MAPLE

G:=z*(1-t*z^2)*(1-z+t*z-z^2-t*z^2-t^2*z^3)/(1-z-t*z^2)/(1-z-z^2-3*t*z^2-t*z^3+t^2*z^4): Gser:=simplify(series(G, z=0, 18)): for n from 1 to 15 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 1 to 15 do seq(coeff(P[n], t, j), j=0..floor(n/2)) od; # yields sequence in triangular form

CROSSREFS

Cf. A001519, A121530, A121531, A054142.

Sequence in context: A059926 A138775 A209385 * A006370 A262370 A108759

Adjacent sequences:  A121526 A121527 A121528 * A121530 A121531 A121532

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Aug 05 2006

STATUS

approved

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Last modified September 23 13:55 EDT 2017. Contains 292358 sequences.