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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=1..n} k*T(n,k) = 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) = t*z*(1-z)^2/((1-3z+z^2)*(1-tz)). From Gary W. Adamson, Jul 07 2011: (Start) Let M be 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. (End) 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

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