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A121461 Triangle read by rows: T(n,k) is the number of nondecreasing Dyck paths of semilength n, having last ascent of length k (1<=k<=n). 2
1, 1, 1, 3, 1, 1, 8, 3, 1, 1, 21, 8, 3, 1, 1, 55, 21, 8, 3, 1, 1, 144, 55, 21, 8, 3, 1, 1, 377, 144, 55, 21, 8, 3, 1, 1, 987, 377, 144, 55, 21, 8, 3, 1, 1, 2584, 987, 377, 144, 55, 21, 8, 3, 1, 1, 6765, 2584, 987, 377, 144, 55, 21, 8, 3, 1, 1, 17711, 6765, 2584, 987, 377, 144, 55, 21 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Also the number of directed column-convex polyominoes of area n, having k cells in the last column. Row sums are the odd-subscripted Fibonacci numbers (A001519). Sum(k*T(n,k),k=1..n)=fibonacci(2n)=A001906(n).

Riordan array ((1-2*x+x^2)/(1-3*x+x^2), x). - Philippe Deléham, Oct 04 2014

Antidiagonal sums are in A007598. - Philippe Deléham, May 22 2015

LINKS

Table of n, a(n) for n=1..74.

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

E. Barcucci, R. Pinzani and R. Sprugnoli, Directed column-convex polyominoes by recurrence relations, Lecture Notes in Computer Science, No. 668, Springer, Berlin (1993), pp. 282-298.

A. M. Baxter, L. K. Pudwell, Ascent sequences avoiding pairs of patterns, 2014.

E. Deutsch and H. Prodinger, A bijection between directed column-convex polyominoes and ordered trees of height at most three, Theoretical Comp. Science, 307, 2003, 319-325.

FORMULA

T(n,k) = fibonacci(2(n-k)) if k<n; T(n,n)=1.

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

Let M = the production matrix:

1, 1, 0, 0, 0, 0,...

2, 0, 1, 0, 0, 0,...

3, 0, 0, 1, 0, 0,...

4, 0, 0, 0, 1, 0,...

5, 0, 0, 0, 0, 1,...

...

n-th row of triangle A121461 =  top row terms of (n-1)-th power of M. - Gary W. Adamson, Jul 07 2011

Let P denote Pascal's triangle. Then P^(-1)*A121461*P = A104762. - Peter Bala, Apr 11 2013

EXAMPLE

T(4,2)=3 because we have UUDD(UU)DD, UUD(UU)DDD and UDUD(UU)DD, where U=(1,1) and D=(1,-1) (the last ascents are shown between parentheses).

Triangle starts:

1;

1,1;

3,1,1;

8,3,1,1;

21,8,3,1,1;

55,21,8,3,1,1;

MAPLE

with(combinat): T:=proc(n, k) if k<n then fibonacci(2*(n-k)) elif k=n then 1 else 0 fi end: for n from 1 to 13 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form

CROSSREFS

Cf. A001519, A001906, A104762, A088305.

Cf. A000045, A007598.

Sequence in context: A131932 A016462 A198618 * A273719 A274488 A203717

Adjacent sequences:  A121458 A121459 A121460 * A121462 A121463 A121464

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Jul 31 2006

STATUS

approved

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Last modified May 12 05:30 EDT 2021. Contains 343812 sequences. (Running on oeis4.)