A331939 Triangle read by rows: Take a pentagon with all diagonals drawn, as in A331929. Then T(n,k) = number of k-sided polygons in that figure for k = 3, 4, ..., n+5.

10, 0, 1, 0, 120, 40, 10, 0, 0, 605, 290, 166, 95, 0, 5, 1750, 1420, 550, 150, 30, 0, 0, 4315, 3740, 1920, 640, 95, 20, 5, 6, 9370, 7950, 3610, 1200, 220, 20, 10, 0, 0, 17290, 15705, 7991, 2885, 520, 75, 20, 5, 0, 0, 29590, 28130, 13560, 4320, 860, 150, 0, 0, 0, 0, 0

Offset: 1

Author

Scott R. Shannon and N. J. A. Sloane, Feb 02 2020

Keywords

• nonn
• tabf

Comments

See the links in A331929 for images of the pentagons.

Examples

			A pentagon with no other points along its edges, n = 1, contains 10 triangles, 1 pentagon and no other n-gons, so the first row is [10,0,1,0]. A pentagon with 1 point dividing its edges, n = 2, contains 120 triangles, 40 quadrilaterals, 10 pentagons and no other n-gons, so the second row is [120, 40, 10, 0, 0].
Triangle begins:
  10,0,1,0
  120,40,10,0,0
  605,290,166,95,0,5
  1750,1420,550,150,30,0,0
  4315,3740,1920,640,95,20,5,6
  9370,7950,3610,1200,220,20,10,0,0
  17290,15705,7991,2885,520,75,20,5,0,0
  29590,28130,13560,4320,860,150,0,0,0,0,0
The row sums are A331929.
		

Links

• Lars Blomberg, Table of n, a(n) for n = 1..735 (the first 35 rows)
• Wikipedia, Pentagon.

Crossrefs

Cf A331929 (regions), A329710 (edges), A330847 (vertices), A331931, A331906, A007678, A092867, A331452.