A121460 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).

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, 233, 145, 94, 65, 43, 22, 7, 1, 610, 378, 239, 159, 108, 65, 29, 8, 1, 1597, 988, 617, 398, 267, 173, 94, 37, 9, 1, 4181, 2585, 1605, 1015, 665, 440, 267, 131, 46, 10, 1, 10946, 6766

Offset: 1

Author

Emeric Deutsch, Jul 31 2006

Keywords

• nonn
• tabl

Comments

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).

Examples

			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).
Triangle starts:
  1;
  1,1;
  2,2,1;
  5,4,3,1;
  13,9,7,4,1;
  34,22,16,11,5,1;
  ...
		

Links

• Elena Barcucci, Alberto del Lungo, S. Fezzi, and Renzo Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.
• Elena Barcucci, Renzo Pinzani, and Renzo Sprugnoli, Directed column-convex polyominoes by recurrence relations, Lecture Notes in Computer Science, No. 668, Springer, Berlin (1993), pp. 282-298.
• Emeric Deutsch and Helmut Prodinger, A bijection between directed column-convex polyominoes and ordered trees of height at most three, Theoretical Comp. Science, 307, 2003, 319-325.
• Rigoberto Flórez, Leandro Junes, and José L. Ramírez, Enumerating several aspects of non-decreasing Dyck paths, Discrete Mathematics (2019) Vol. 342, Issue 11, 3079-3097. See page 3087.
• Juan B. Gil, Felix H. Xu, and William Y. Zhu, Odd-indexed Fibonacci numbers via pattern-avoiding permutations, arXiv:2506.15800 [math.CO], 2025. See p. 7.

Crossrefs

Cf. A001519, A055588, A061667.

Programs

Maple

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

Formula

T(n,k) = binomial(n-2,k-2)+Sum(fibonacci(2j-1)*binomial(n-2-j,k-2), j=1..n-k).

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