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Search: id:A134436
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| A134436 |
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Triangle read by rows: T(n,k) is the number of deco polyominoes of height n with k cells in the second row (0<=k<=n-1; 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, 1, 1, 1, 3, 2, 1, 7, 10, 6, 1, 16, 37, 42, 24, 1, 44, 125, 214, 216, 120, 1, 169, 465, 959, 1406, 1320, 720, 1, 895, 2199, 4469, 7880, 10476, 9360, 5040, 1, 5942, 13504, 24902, 44203, 70676, 87732, 75600, 40320
(list; table; graph; listen)
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OFFSET
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1,5
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COMMENT
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Row sums are the factorials (A000142). Sum(k*T(n,k),k=0..n-1)=A134437(n).
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REFERENCES
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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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The generating polynomial of row n is P(n,t)=Q(n,t,1), where Q(1,t,x)=x and Q(n,t,x)=xQ(n-1,1,t)+x(1+(n-2)t)Q(n-1,t,x) for n>=2.
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EXAMPLE
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T(2,0)=T(2,1)=1 because the horizontal domino has no cells in the 2nd row and the vertical domino has 1 cell in the 2nd row.
Triangle starts:
1;
1,1;
1,3,2;
1,7,10,6;
1,16,37,42,24;
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MAPLE
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Q[1]:=x: for n from 2 to 10 do Q[n]:=expand(x*subs({t=1, x=t}, Q[n-1])+x*Q[n-1]+(n-2)*t*x*Q[n-1]) end do: for n to 9 do P[n]:=sort(subs(x=1, Q[n])) end do: for n to 9 do seq(coeff(P[n], t, j), j=0..n-1) end do; # yields sequence in triangular form
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CROSSREFS
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Cf. A000142, A134437.
Adjacent sequences: A134433 A134434 A134435 this_sequence A134437 A134438 A134439
Sequence in context: A145035 A122832 A056151 this_sequence A028246 A082038 A143774
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 30 2007
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