A121298 - OEIS

A121298

Triangle read by rows: T(n,k) is the number of directed column-convex polyominoes of area n and height k (1<=k<=n; here by the height of a polyomino one means the number of lines of slope -1 that pass through the centers of the polyomino cells).

1, 0, 2, 0, 1, 4, 0, 0, 5, 8, 0, 0, 3, 15, 16, 0, 0, 1, 17, 39, 32, 0, 0, 0, 15, 59, 95, 64, 0, 0, 0, 9, 75, 175, 223, 128, 0, 0, 0, 4, 78, 269, 479, 511, 256, 0, 0, 0, 1, 67, 358, 845, 1247, 1151, 512, 0, 0, 0, 0, 48, 419, 1300, 2461, 3135, 2559, 1024, 0, 0, 0, 0, 29, 432, 1801, 4224, 6813, 7679, 5631, 2048

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

OFFSET

1,3

COMMENTS

Row sums are the odd-subscripted Fibonacci numbers (A001519). Sum of terms in column k = A007808(k). Sum(k*T(n,k),k=0..n)=A121299(n).

LINKS

Table of n, a(n) for n=1..78.

E. Barcucci, A. Del Lungo, R. Pinzani and R. Sprugnoli, La hauteur des polyominos dirigés verticalement convexes, Actes du 31e Séminaire Lotharingien de Combinatoire, Publi. IRMA, Université Strasbourg I (1993).

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.

FORMULA

T(n,k) = T(n-1,k-1)+Sum(T(n-k,j), j=1..k-1)+Sum(T(n-j,k-1), j=1..k-1).

EXAMPLE

T(2,2)=2 because we have the vertical and the horizontal dominoes.

Triangle starts:

0,2;

0,1,4;

0,0,5,8;

0,0,3,15,16;

0,0,1,17,39,32;

MAPLE

T:=proc(n, k) if n<=0 or k<=0 then 0 elif n=1 and k=1 then 1 else T(n-1, k-1)+add(T(n-k, j), j=1..k-1)+add(T(n-j, k-1), j=1..k-1) fi end: for n from 1 to 12 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form

MATHEMATICA

T[n_, k_] := T[n, k] = Which[n <= 0 || k <= 0, 0, n == 1 && k == 1, 1, True, T[n-1, k-1] + Sum[T[n-k, j], {j, 1, k-1}] + Sum[T[n-j, k-1], {j, 1, k-1}]];

Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 25 2024, after Maple program *)

CROSSREFS

Cf. A001519, A007808, A121299.

Sequence in context: A061290 A099096 A099089 * A372873 A212206 A247489

Adjacent sequences: A121295 A121296 A121297 * A121299 A121300 A121301

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Aug 04 2006

STATUS

approved