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

Offset: 1

Author

Emeric Deutsch, Aug 04 2006

Keywords

• nonn

Examples

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

Links

• 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.

Crossrefs

Cf. A121298.

Programs

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 *)

Formula

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

Extensions

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