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A121694 Sum of the vertical heights (i.e., number of rows) of all deco polyominoes of height n.

%I A121694 #16 Aug 20 2024 11:27:40
%S A121694 1,3,12,61,377,2734,22671,211035,2175754,24592551,302295925,
%T A121694 4014475756,57277225309,873819665135,14195291340656,244657733062761,
%U A121694 4459137940238245,85694418205589534,1731893273528613811,36721566227335477047,815098440677104096866
%N A121694 Sum of the vertical heights (i.e., number of rows) of all deco polyominoes of height n.
%C A121694 A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.
%H A121694 E. Barcucci, S. Brunetti, and F. Del Ristoro, <a href="http://www.numdam.org/item?id=ITA_2000__34_1_1_0">Succession rules and deco polyominoes</a>, Theoret. Informatics Appl., 34, 2000, 1-14.
%H A121694 E. Barcucci, A. Del Lungo, and R. Pinzani, <a href="http://dx.doi.org/10.1016/0304-3975(95)00199-9">"Deco" polyominoes, permutations and random generation</a>, Theoretical Computer Science, 159, 1996, 29-42.
%F A121694 a(n) = Sum_{k=1..n} k*A121692(n,k).
%F A121694 a(n) = Sum_{k=1..n} k*T(n,k), where T(n,k) (A121692) is defined by T(n,1)=1; T(n,n)=1; T(n,k) = k*T(n-1,k) + 2*T(n-1,k-1) + Sum_{j=1..k-2} T(n-1,j) for k <= n; T(n,k)=0 for k > n.
%e A121694 a(2)=3 because the deco polyominoes of height 2 are the horizontal and vertical dominoes, having, respectively, 1 and 2 rows.
%p A121694 with(linalg): a:=proc(i,j) if i=j then i elif i>j then 1 else 0 fi end: p:=proc(Q) local n,A,b,w,QQ: n:=degree(Q): A:=matrix(n,n,a): b:=j->coeff(Q,t,j): w:=matrix(n,1,b): QQ:=multiply(A,w): sort(expand(add(QQ[k,1]*t^k,k=1..n)+t*Q)): end: P[1]:=t: for n from 2 to 22 do P[n]:=p(P[n-1]) od: seq(subs(t=1,diff(P[n],t)),n=1..22);
%t A121694 (* T is A121692 *)
%t A121694 T[n_, k_] := T[n, k] = Which[k == 1, 1, k == n, 1, k > n, 0, True, k*T[n-1, k] + 2*T[n-1, k-1] + Sum[T[n-1, j], {j, 1, k-2}]];
%t A121694 a[n_] := Sum[k*T[n, k], {k, 1, n}];
%t A121694 Table[a[n], {n, 1, 21}] (* _Jean-François Alcover_, Aug 20 2024 *)
%Y A121694 Cf. A121692.
%K A121694 nonn
%O A121694 1,2
%A A121694 _Emeric Deutsch_, Aug 17 2006