Search: a121636 -id:a121636
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A121634
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Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and having k 2-cell columns starting at level 0 (n >= 1; 0 <= k <= n-1).
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+10
2
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1, 1, 1, 2, 3, 1, 8, 10, 5, 1, 42, 44, 25, 8, 1, 264, 242, 144, 57, 12, 1, 1920, 1594, 962, 429, 117, 17, 1, 15840, 12204, 7366, 3536, 1131, 219, 23, 1, 146160, 106308, 63766, 32118, 11453, 2664, 380, 30, 1, 1491840, 1036944, 616436, 320710, 123742, 32765, 5704, 620, 38, 1
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OFFSET
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1,4
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COMMENTS
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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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LINKS
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FORMULA
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Row sums are the factorials (A000142).
Sum_{k=0..n-1} k*T(n,k) = A121636(n).
The row generating polynomials satisfy P(n,t) = (t+n-2)[(n-2)!+P(n-1,t)] for n >= 3, P(1,t)=1 and P(2,t)=1+t.
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EXAMPLE
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T(2,0)=1 and T(2,1)=1 because the deco polyominoes of height 2 are the horizontal and vertical dominoes, having, respectively, 0 and 1 columns with exactly 2 cells starting at level 0.
Triangle starts:
1;
1, 1;
2, 3, 1;
8, 10, 5, 1;
42, 44, 25, 8, 1;
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MAPLE
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P[1]:=1: P[2]:=1+t: for n from 3 to 11 do P[n]:=sort(expand((t+n-2)*((n-2)!+P[n-1]))) od: for n from 1 to 11 do seq(coeff(P[n], t, j), j=0..n-1) od; # yields sequence in triangular form
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MATHEMATICA
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P[n_ /; n >= 3, t_] := P[n, t] = (t + n - 2) ((n - 2)! + P[n - 1, t]);
P[1, _] = 1; P[2, t_] = 1 + t;
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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