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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, 36721566227335477047, 815098440677104096866
OFFSET
1,2
COMMENTS
A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.
LINKS
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.
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(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.
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[n-1]) od: seq(subs(t=1, diff(P[n], t)), n=1..22);
MATHEMATICA
(* T is A121692 *)
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}]];
a[n_] := Sum[k*T[n, k], {k, 1, n}];
Table[a[n], {n, 1, 21}] (* Jean-François Alcover, Aug 20 2024 *)
CROSSREFS
Cf. A121692.
Sequence in context: A331607 A235802 A317169 * A331616 A158691 A365299
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Aug 17 2006
STATUS
approved