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%I #7 Jul 22 2017 08:35:35
%S 1,3,12,61,377,2734,22671,211035,2175754,24592551,302295925,
%T 4014475756,57277225309,873819665135,14195291340656,244657733062761,
%U 4459137940238245,85694418205589534,1731893273528613811
%N Sum of the vertical heights (i.e., number of rows) of all deco polyominoes of height n.
%C 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 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 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 a(n) = Sum_{k=1..n} k*A121692(n,k).
%F 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 a(2)=3 because the deco polyominoes of height 2 are the horizontal and vertical dominoes, having, respectively, 1 and 2 rows.
%p 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);
%Y Cf. A121692.
%K nonn
%O 1,2
%A _Emeric Deutsch_, Aug 17 2006
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