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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. 0
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

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.

REFERENCES

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.

LINKS

Table of n, a(n) for n=1..68.

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

FORMULA

T(n, k)=k*fibonacci(2n-2k-1) if k<n; T(n, n)=1. G.f.=tz(1-2z-tz+3tz^2-tz^3)/[(1-3z+z^2)(1-tz)^2].

EXAMPLE

Triangle begins:

1;

1,1;

2,2,1;

5,4,3,1;

13,10,6,4,1;

MAPLE

with(combinat):T:=proc(n, k) if k<n then k*fibonacci(2*n-2*k-1) elif k=n then 1 else 0 fi end:for n from 1 to 12 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form

MATHEMATICA

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

CROSSREFS

Cf. A001519, A055819.

Sequence in context: A110438 A184051 A121460 * A273342 A276067 A125177

Adjacent sequences:  A105289 A105290 A105291 * A105293 A105294 A105295

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Apr 25 2005

STATUS

approved

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Last modified March 23 18:13 EDT 2019. Contains 321433 sequences. (Running on oeis4.)