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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).
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%I #32 Feb 14 2022 03:55:13

%S 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,

%T 144,55,21,8,3,1,1,987,377,144,55,21,8,3,1,1,2584,987,377,144,55,21,8,

%U 3,1,1,6765,2584,987,377,144,55,21,8,3,1,1,17711,6765,2584,987,377,144,55,21

%N 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).

%C 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).

%C Riordan array ((1-2*x+x^2)/(1-3*x+x^2), x). - _Philippe Deléham_, Oct 04 2014

%C Antidiagonal sums are in A007598. - _Philippe Deléham_, May 22 2015

%H E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, <a href="http://dx.doi.org/10.1016/S0012-365X(97)82778-1">Nondecreasing Dyck paths and q-Fibonacci numbers</a>, Discrete Math., 170, 1997, 211-217.

%H E. Barcucci, R. Pinzani and R. Sprugnoli, <a href="http://dx.doi.org/10.1007/3-540-56610-4_71">Directed column-convex polyominoes by recurrence relations</a>, Lecture Notes in Computer Science, No. 668, Springer, Berlin (1993), pp. 282-298.

%H A. M. Baxter, L. K. Pudwell, <a href="http://faculty.valpo.edu/lpudwell/papers/AvoidingPairs.pdf">Ascent sequences avoiding pairs of patterns</a>, 2014.

%H E. Deutsch and H. Prodinger, <a href="http://dx.doi.org/10.1016/S0304-3975(03)00222-6">A bijection between directed column-convex polyominoes and ordered trees of height at most three</a>, Theoretical Comp. Science, 307, 2003, 319-325.

%F T(n,k) = Fibonacci(2(n-k)) if k < n; T(n,n)=1.

%F G.f.: G = G(t,z) = t*z*(1-z)^2/((1-3z+z^2)*(1-tz)).

%F From _Gary W. Adamson_, Jul 07 2011: (Start)

%F Let M be the production matrix:

%F 1, 1, 0, 0, 0, 0, ...

%F 2, 0, 1, 0, 0, 0, ...

%F 3, 0, 0, 1, 0, 0, ...

%F 4, 0, 0, 0, 1, 0, ...

%F 5, 0, 0, 0, 0, 1, ...

%F ...

%F n-th row of triangle A121461 = top row terms of (n-1)-th power of M. (End)

%F Let P denote Pascal's triangle. Then P^(-1)*A121461*P = A104762. - _Peter Bala_, Apr 11 2013

%e 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).

%e Triangle starts:

%e 1;

%e 1, 1;

%e 3, 1, 1;

%e 8, 3, 1, 1;

%e 21, 8, 3, 1, 1;

%e 55, 21, 8, 3, 1, 1;

%e ...

%p 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

%Y Cf. A001519, A001906, A104762, A088305.

%Y Cf. A000045, A007598.

%K nonn,tabl

%O 1,4

%A _Emeric Deutsch_, Jul 31 2006