A121299 - OEIS

A121299

Sum of the heights of all directed column-convex polyominoes of area 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, 4, 14, 47, 149, 458, 1373, 4046, 11765, 33857, 96611, 273760, 771164, 2161352, 6031104, 16764719, 46442640, 128268379, 353296944, 970717966, 2661204271, 7280832780, 19882745230, 54203791062, 147536291969, 400991600305

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

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..26.

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

a(n) = Sum_{k=1..n} k*A121298(n,k). [Corrected by R. J. Mathar, Sep 18 2007]

EXAMPLE

a(2)=4 because the vertical and the horizontal dominoes have altogether 4 diagonals with slope -1.

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: seq(add(k*T(n, k), k=1..n), n=1..15);

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}]];

a[n_] := Sum[k*T[n, k], {k, 1, n}];

Table[a[n], {n, 1, 26}] (* Jean-François Alcover, Aug 25 2024, after Maple program *)

CROSSREFS

Cf. A121298.

Sequence in context: A124805 A362279 A121530 * A326346 A046718 A291385

Adjacent sequences: A121296 A121297 A121298 * A121300 A121301 A121302

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Aug 04 2006

EXTENSIONS

More terms from R. J. Mathar, Sep 18 2007

STATUS

approved