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%I #23 Dec 26 2017 23:31:54
%S 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,
%T 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,
%U 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
%N Triangle of the number of nondegenerate k-gons having perimeter n and whose sides are nondecreasing, for k=3..n.
%C For a k-gon to be nondegenerate, the longest side must be shorter than the sum of the remaining sides.
%C 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
%H T. D. Noe, <a href="/A124278/b124278.txt">Rows n=3..102 of triangle, flattened</a>
%H G. E. Andrews, P. Paule and A. Riese, <a href="http://www.risc.uni-linz.ac.at/publications/download/risc_163/PAIX.pdf">MacMahon's Partition Analysis IX: k-Gon Partitions</a>, Bull. Austral. Math. Soc. 64 (2001), 321-329.
%H James East, Ron Niles, <a href="https://arxiv.org/abs/1710.11245">Integer polygons of given perimeter</a>, arXiv:1710.11245 [math.CO], 2017.
%F 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)).
%e For polygons having perimeter 7: 2 triangles, 2 quadrilaterals, 2 pentagons, 1 hexagon and 1 heptagon. The triangle begins
%e 1
%e 0 1
%e 1 1 1
%e 1 1 1 1
%e 2 2 2 1 1
%e 1 3 2 2 1 1
%p b:= proc(n, i) option remember; `if`(n=0, [1], `if`(i<1, [],
%p zip((x, y)-> x+y, b(n, i-1), `if`(i>n, [],
%p [0, b(n-i, i)[]]), 0)))
%p end:
%p T:= n-> b(n, ceil(n/2)-1)[4..n+1][]:
%p seq(T(n), n=3..20); # _Alois P. Heinz_, Jul 15 2013
%t Flatten[Table[p=IntegerPartitions[n]; Length[Select[p, Length[ # ]==k && #[[1]] < Total[Rest[ # ]]&]], {n,3,30}, {k,3,n}]]
%t (* second program: *)
%t 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 *)
%Y Cf. A124287 (similar, but with no restriction on the sides).
%Y Cf. A210249 (gives row sums of this sequence for n >= 3).
%Y Cf. A005044 (k=3), A062890 (k=4), A069906 (k=5), A069907 (k=6), A288253 (k=7), A288254 (k=8), A288255 (k=9), A288256 (k=10).
%K nice,nonn,tabl
%O 3,11
%A _T. D. Noe_, Oct 24 2006
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