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a(n) = Sum(_{k=1..n} k*A121298(n,k), k=1..n). [Corrected by R. J. Mathar, Sep 18 2007]
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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 *)
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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).
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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...</a>