A105292 Triangle read by rows: T(n,k) is the number of directed column-convex polyominoes of area n, having leftmost column of height k.

1, 1, 1, 2, 2, 1, 5, 4, 3, 1, 13, 10, 6, 4, 1, 34, 26, 15, 8, 5, 1, 89, 68, 39, 20, 10, 6, 1, 233, 178, 102, 52, 25, 12, 7, 1, 610, 466, 267, 136, 65, 30, 14, 8, 1, 1597, 1220, 699, 356, 170, 78, 35, 16, 9, 1, 4181, 3194, 1830, 932, 445, 204, 91, 40, 18, 10, 1, 10946, 8362

Offset: 1

Author

Emeric Deutsch, Apr 25 2005

Keywords

• nonn
• tabl

Comments

T(n,k) is the number of nondecreasing Dyck paths of semilength n, having height of leftmost peak equal to k. Example: T(3,2)=2 because we have UUDDUD and UUDUDD, where U=(1,1) and D(1,-1). Sum of row n = fibonacci(2n-1) (A001519). T(n,1)=fibonacci(2n-3) (A001519). Column 2 yields A055819.

Examples

			Triangle begins:
  1;
  1,1;
  2,2,1;
  5,4,3,1;
  13,10,6,4,1;
		

Links

• Elena Barcucci, Alberto del Lungo, S. Fezzi, and Renzo Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.
• 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.
• 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. 4.

Crossrefs

Cf. A001519, A055819.

Programs

Maple

with(combinat):T:=proc(n,k) if k
				

Mathematica

Flatten[Join[{1},#]&/@Table[k*Fibonacci[2n-2k-1],{n,15},{k,n-1}]] (* Harvey P. Dale, Aug 21 2013 *)

Formula

T(n, k)=k*fibonacci(2n-2k-1) if k

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.