

A121694


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


1



1, 3, 12, 61, 377, 2734, 22671, 211035, 2175754, 24592551, 302295925, 4014475756, 57277225309, 873819665135, 14195291340656, 244657733062761, 4459137940238245, 85694418205589534, 1731893273528613811
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OFFSET

1,2


COMMENTS

A deco polyomino is a directed columnconvex polyomino in which the height, measured along the diagonal, is attained only in the last column.


LINKS

Table of n, a(n) for n=1..19.
E. Barcucci, S. Brunetti and F. Del Ristoro, Succession rules and deco polyominoes, Theoret. Informatics Appl., 34, 2000, 114.
E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 2942.


FORMULA

a(n) = Sum_{k=1..n} k*A121692(n,k).
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(n1,k) + 2*T(n1,k1) + Sum_{j=1..k2} T(n1,j) for k <= n; T(n,k)=0 for k > n.


EXAMPLE

a(2)=3 because the deco polyominoes of height 2 are the horizontal and vertical dominoes, having, respectively, 1 and 2 rows.


MAPLE

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[n1]) od: seq(subs(t=1, diff(P[n], t)), n=1..22);


CROSSREFS

Cf. A121692.
Sequence in context: A331607 A235802 A317169 * A331616 A158691 A038171
Adjacent sequences: A121691 A121692 A121693 * A121695 A121696 A121697


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Aug 17 2006


STATUS

approved



