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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 (list; graph; refs; listen; history; text; internal format)



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


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, 1-14.

E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.


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(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.


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


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);


Cf. A121692.

Sequence in context: A331607 A235802 A317169 * A331616 A158691 A038171

Adjacent sequences:  A121691 A121692 A121693 * A121695 A121696 A121697




Emeric Deutsch, Aug 17 2006



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Last modified December 2 05:03 EST 2020. Contains 338865 sequences. (Running on oeis4.)