1,4

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).

E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.

E. Barcucci, R. Pinzani and R. Sprugnoli, Directed column-convex polyominoes by recurrence relations, Lecture Notes in Computer Science, No. 668, Springer, Berlin (1993), pp. 282-298.

E. Deutsch and H. Prodinger, A bijection between directed column-convex polyominoes and ordered trees of height at most three, Theoretical Comp. Science, 307, 2003, 319-325.

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

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;

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

Cf. A001519, A055588, A061667.

Sequence in context: A134226 A127742 A110438 this_sequence A105292 A125177 A125178

Adjacent sequences: A121457 A121458 A121459 this_sequence A121461 A121462 A121463

nonn,tabl

Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 31 2006