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Search: id:A121694
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| A121694 |
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Sum of the vertical heights (i.e. number of rows) of all deco polyominoes of height n. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column. |
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+0
2
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| 1, 3, 12, 61, 377, 2734, 22671, 211035, 2175754, 24592551, 302295925, 4014475756, 57277225309, 873819665135, 14195291340656, 244657733062761, 4459137940238245, 85694418205589534, 1731893273528613811
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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a(n)=Sum(k*A121692(n,k),k=1..n).
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REFERENCES
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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.
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FORMULA
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a(n)=Sum(k*T(n,k),k=1..n), 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(T(n-1,j), j=1..k-2) for k<=n; T(n,k)=0 for k>n.
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EXAMPLE
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a(2)=3 because the deco polyominoes of height 2 are the horizontal and vertical dominoes, having, respectively, 1 and 2 rows.
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MAPLE
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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);
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CROSSREFS
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Cf. A121692.
Sequence in context: A082278 A078162 A002497 this_sequence A038171 A074516 A045740
Adjacent sequences: A121691 A121692 A121693 this_sequence A121695 A121696 A121697
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 17 2006
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