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A121301
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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 shortest column (1<=k<=n).
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1
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1, 1, 1, 4, 0, 1, 10, 2, 0, 1, 28, 5, 0, 0, 1, 75, 10, 3, 0, 0, 1, 202, 23, 7, 0, 0, 0, 1, 540, 57, 8, 4, 0, 0, 0, 1, 1440, 129, 18, 9, 0, 0, 0, 0, 1, 3828, 294, 43, 10, 5, 0, 0, 0, 0, 1, 10153, 680, 90, 11, 11, 0, 0, 0, 0, 0, 1, 26875, 1557, 178, 28, 12, 6, 0, 0, 0, 0, 0, 1, 71021, 3546
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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COMMENTS
| Row sums are the odd-subscripted Fibonacci numbers (A001519). T(n,1)=fibonacci(2n-1)-A121469(n,0) (obviously, since A121469(n,k) is the number of directed column-convex polyominoes of area n and having k 1-cell columns). T(n,n)=1.
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REFERENCES
| 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
| G.f. of column k is f[k]-f[k+1], where f[k]=z^k*(1-z)/(z^2-2*z+1+z^(1+k)*k-k*z^k-z^(1+k)) is the g.f. for directed column-convex polyominoes whose columns have height at least k.
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EXAMPLE
| Triangle starts:
1;
1,1;
4,0,1;
10,2,0,1;
28,5,0,0,1;
75,10,3,0,0,1;
202,23,7,0,0,0,1;
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MAPLE
| f:=k->z^k*(1-z)/(z^2-2*z+1+z^(1+k)*k-k*z^k-z^(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 13 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form
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CROSSREFS
| Cf. A001519, A121469, A121300.
Sequence in context: A186761 A199786 A189245 * A059056 A127153 A178979
Adjacent sequences: A121298 A121299 A121300 * A121302 A121303 A121304
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KEYWORD
| nonn,tabl
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 04 2006
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