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Search: id:A121300
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| A121300 |
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Triangle read by rows: T(n,k) is the number of directed column-convex polyominoes of area n and having k cells in the longest column (1<=k<=n). |
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+0
2
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| 1, 1, 1, 1, 3, 1, 1, 7, 4, 1, 1, 15, 12, 5, 1, 1, 31, 35, 15, 6, 1, 1, 63, 95, 48, 18, 7, 1, 1, 127, 249, 145, 58, 21, 8, 1, 1, 255, 640, 418, 181, 68, 24, 9, 1, 1, 511, 1615, 1180, 545, 213, 78, 27, 10, 1, 1, 1023, 4026, 3279, 1593, 649, 245, 88, 30, 11, 1, 1, 2047, 9944, 8981
(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 odd-subscripted Fibonacci numbers (A001519). T(n,1)=1; T(n,2)=2^(n-1)-1=A000225(n-1); T(n,n)=1.
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REFERENCES
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E. Barcucci, R. Pinzani and R. Sprugnoli, Directed column-convex polyominoes by recurrence relations, Lecture Notes in Computer Science, No. 668, Springer, Berlin (1993), pp. 282-298.
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FORMULA
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G.f. of column k is f[k]-f[k-1], where f[k]=Sum(z^i, i=1..k)/[1-Sum(jz^j, j=1..k)] is the g.f. for directed column-convex polyominoes whose columns have height at most k.
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EXAMPLE
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Triangle starts:
1;
1,1;
1,3,1;
1,7,4,1;
1,15,12,5,1;
1,31,35,15,6,1;
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MAPLE
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f:=k->sum(z^i, i=1..k)/(1-sum(j*z^j, j=1..k)): T:=proc(n, k) if k<=n then coeff(series(f(k)-f(k-1), z=0, 15), z, n) else 0 fi end: for n from 1 to 12 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A001519, A000225.
Sequence in context: A078026 A126713 A140068 this_sequence A128119 A112996 A136621
Adjacent sequences: A121297 A121298 A121299 this_sequence A121301 A121302 A121303
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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), Aug 04 2006
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