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A124278
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Triangle of the number of nondegenerate k-gons having perimeter n and whose sides are nondecreasing, for k=3..n.
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10
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1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 3, 2, 2, 1, 1, 3, 4, 4, 3, 2, 1, 1, 2, 5, 5, 4, 3, 2, 1, 1, 4, 7, 8, 6, 5, 3, 2, 1, 1, 3, 8, 9, 9, 6, 5, 3, 2, 1, 1, 5, 11, 14, 12, 10, 7, 5, 3, 2, 1, 1, 4, 12, 16, 16, 13, 10, 7, 5, 3, 2, 1, 1, 7, 16, 23, 22, 19, 14, 11, 7, 5, 3, 2, 1, 1, 5, 18, 25, 28, 24, 20, 14, 11, 7, 5, 3, 2, 1, 1
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OFFSET
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3,11
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COMMENTS
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For a k-gon to be nondegenerate, the longest side must be shorter than the sum of the remaining sides.
T(n,k) = number of partitions of n into k parts (k >= 3) in which all parts are less than n/2. Also T(n,k) = number of partitions of 2*n into k parts in which all parts are even and less than n. - L. Edson Jeffery, Mar 19 2012
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LINKS
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FORMULA
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G.f. for column k is x^k/(product_{i=1..k} 1-x^i) - x^(2k-2)/(1-x)/(product_{i=1..k-1} 1-x^(2i)).
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EXAMPLE
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For polygons having perimeter 7: 2 triangles, 2 quadrilaterals, 2 pentagons, 1 hexagon and 1 heptagon. The triangle begins
1
0 1
1 1 1
1 1 1 1
2 2 2 1 1
1 3 2 2 1 1
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0, [1], `if`(i<1, [],
zip((x, y)-> x+y, b(n, i-1), `if`(i>n, [],
[0, b(n-i, i)[]]), 0)))
end:
T:= n-> b(n, ceil(n/2)-1)[4..n+1][]:
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MATHEMATICA
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Flatten[Table[p=IntegerPartitions[n]; Length[Select[p, Length[ # ]==k && #[[1]] < Total[Rest[ # ]]&]], {n, 3, 30}, {k, 3, n}]]
(* second program: *)
QP = QPochhammer; T[n_, k_] := SeriesCoefficient[x^k*(1/QP[x, x, k] + x^(k - 2)/((x-1)*QP[x^2, x^2, k-1])), {x, 0, n}]; Table[T[n, k], {n, 3, 16}, {k, 3, n}] // Flatten (* Jean-François Alcover, Jan 08 2016 *)
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CROSSREFS
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Cf. A124287 (similar, but with no restriction on the sides).
Cf. A210249 (gives row sums of this sequence for n >= 3).
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KEYWORD
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AUTHOR
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STATUS
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approved
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