A124278 Triangle of the number of nondegenerate k-gons having perimeter n and whose sides are nondecreasing, for k=3..n.

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

Offset: 3

Author

T. D. Noe, Oct 24 2006

Keywords

• nice
• nonn
• tabl

Comments

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

Examples

			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
		

Links

• T. D. Noe, Rows n=3..102 of triangle, flattened
• G. E. Andrews, P. Paule and A. Riese, MacMahon's Partition Analysis IX: k-Gon Partitions, Bull. Austral. Math. Soc. 64 (2001), 321-329.
• James East, Ron Niles, Integer polygons of given perimeter, arXiv:1710.11245 [math.CO], 2017.

Crossrefs

Cf. A124287 (similar, but with no restriction on the sides).

Cf. A210249 (gives row sums of this sequence for n >= 3).

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).

Programs

Maple

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][]:
seq(T(n), n=3..20);  # Alois P. Heinz, Jul 15 2013

Mathematica

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 *)

Formula

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)).