This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A121461 #32 Feb 14 2022 03:55:13
%S A121461 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 A121461 144,55,21,8,3,1,1,987,377,144,55,21,8,3,1,1,2584,987,377,144,55,21,8,
%U A121461 3,1,1,6765,2584,987,377,144,55,21,8,3,1,1,17711,6765,2584,987,377,144,55,21
%N 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).
%C A121461 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 A121461 Riordan array ((1-2*x+x^2)/(1-3*x+x^2), x). - _Philippe Deléham_, Oct 04 2014
%C A121461 Antidiagonal sums are in A007598. - _Philippe Deléham_, May 22 2015
%H A121461 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 A121461 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 A121461 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 A121461 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 A121461 T(n,k) = Fibonacci(2(n-k)) if k < n; T(n,n)=1.
%F A121461 G.f.: G = G(t,z) = t*z*(1-z)^2/((1-3z+z^2)*(1-tz)).
%F A121461 From _Gary W. Adamson_, Jul 07 2011: (Start)
%F A121461 Let M be the production matrix:
%F A121461 1, 1, 0, 0, 0, 0, ...
%F A121461 2, 0, 1, 0, 0, 0, ...
%F A121461 3, 0, 0, 1, 0, 0, ...
%F A121461 4, 0, 0, 0, 1, 0, ...
%F A121461 5, 0, 0, 0, 0, 1, ...
%F A121461 ...
%F A121461 n-th row of triangle A121461 = top row terms of (n-1)-th power of M. (End)
%F A121461 Let P denote Pascal's triangle. Then P^(-1)*A121461*P = A104762. - _Peter Bala_, Apr 11 2013
%e A121461 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 A121461 Triangle starts:
%e A121461 1;
%e A121461 1, 1;
%e A121461 3, 1, 1;
%e A121461 8, 3, 1, 1;
%e A121461 21, 8, 3, 1, 1;
%e A121461 55, 21, 8, 3, 1, 1;
%e A121461 ...
%p A121461 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 A121461 Cf. A001519, A001906, A104762, A088305.
%Y A121461 Cf. A000045, A007598.
%K A121461 nonn,tabl
%O A121461 1,4
%A A121461 _Emeric Deutsch_, Jul 31 2006