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%I #12 Jan 07 2024 10:53:43
%S 1,1,1,2,2,1,5,4,3,1,13,9,7,4,1,34,22,16,11,5,1,89,56,38,27,16,6,1,
%T 233,145,94,65,43,22,7,1,610,378,239,159,108,65,29,8,1,1597,988,617,
%U 398,267,173,94,37,9,1,4181,2585,1605,1015,665,440,267,131,46,10,1,10946,6766
%N Triangle read by rows: T(n,k) is the number of nondecreasing Dyck paths of semilength n, having k returns to the x-axis (1<=k<=n).
%C Also the number of directed column-convex polyominoes of area n, having k cells in the bottom row. Row sums are the odd-subscripted Fibonacci numbers (A001519). T(n,1)=fibonacci(2n-3) for n>=2 (A001519). T(n,2)=1+fibonacci(2n-4)=A055588(n-2). T(n,3)=n-3+fibonacci(2n-5). Sum(k*T(n,k),k=1..n)=A061667(n-1).
%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 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.
%H Rigoberto Flórez, Leandro Junes, and José L. Ramírez, <a href="https://doi.org/10.1016/j.disc.2019.06.018">Enumerating several aspects of non-decreasing Dyck paths</a>, Discrete Mathematics (2019) Vol. 342, Issue 11, 3079-3097. See page 3087.
%F T(n,k) = binomial(n-2,k-2)+Sum(fibonacci(2j-1)*binomial(n-2-j,k-2), j=1..n-k).
%F G.f.: G(t,z)=tz(1-2z)(1-z)/[(1-3z+z^2)(1-z-tz)].
%e T(4,2)=4 because we have UUDDUUDD, UDUUUDDD, UUUDDDUD and UDUUDUDD, where U=(1,1) and D=(1,-1) (the Dyck path UUDUDDUD does not qualify: it does have 2 returns to the x-axis but it is not nondecreasing since its valleys are at altitudes 1 and 0).
%e Triangle starts:
%e 1;
%e 1,1;
%e 2,2,1;
%e 5,4,3,1;
%e 13,9,7,4,1;
%e 34,22,16,11,5,1;
%e ...
%p with(combinat): T:=(n,k)->binomial(n-2,k-2)+add(fibonacci(2*j-1)*binomial(n-2-j,k-2),j=1..n-k): for n from 1 to 12 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form
%Y Cf. A001519, A055588, A061667.
%K nonn,tabl
%O 1,4
%A _Emeric Deutsch_, Jul 31 2006
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