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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.

%I A121299 #17 Aug 25 2024 13:36:08
%S A121299 1,4,14,47,149,458,1373,4046,11765,33857,96611,273760,771164,2161352,
%T A121299 6031104,16764719,46442640,128268379,353296944,970717966,2661204271,
%U A121299 7280832780,19882745230,54203791062,147536291969,400991600305
%N 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.
%H A121299 E. Barcucci, A. Del Lungo, R. Pinzani and R. Sprugnoli, <a href="http://www.mat.univie.ac.at/~slc/opapers/s31barc.html">La hauteur des polyominos dirigés verticalement convexes</a>, Actes du 31e Séminaire Lotharingien de Combinatoire, Publi. IRMA, Université Strasbourg I (1993).
%H A121299 E. Barcucci, R. Pinzani and R. Sprugnoli, <a href="http://dx.doi.org/10.1007/3-540-56610-4_71">Directed column-convex polyominoes by recurrence relations</a>, Lecture Notes in Computer Science, No. 668, Springer, Berlin (1993), pp. 282-298.
%F A121299 a(n) = Sum_{k=1..n} k*A121298(n,k). [Corrected by _R. J. Mathar_, Sep 18 2007]
%e A121299 a(2)=4 because the vertical and the horizontal dominoes have altogether 4 diagonals with slope -1.
%p A121299 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);
%t A121299 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}]];
%t A121299 a[n_] := Sum[k*T[n, k], {k, 1, n}];
%t A121299 Table[a[n], {n, 1, 26}] (* _Jean-François Alcover_, Aug 25 2024, after Maple program *)
%Y A121299 Cf. A121298.
%K A121299 nonn
%O A121299 1,2
%A A121299 _Emeric Deutsch_, Aug 04 2006
%E A121299 More terms from _R. J. Mathar_, Sep 18 2007