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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.

%I A331939 #17 May 11 2020 11:18:36
%S A331939 10,0,1,0,120,40,10,0,0,605,290,166,95,0,5,1750,1420,550,150,30,0,0,
%T A331939 4315,3740,1920,640,95,20,5,6,9370,7950,3610,1200,220,20,10,0,0,17290,
%U A331939 15705,7991,2885,520,75,20,5,0,0,29590,28130,13560,4320,860,150,0,0,0,0,0
%N 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.
%C A331939 See the links in A331929 for images of the pentagons.
%H A331939 Lars Blomberg, <a href="/A331939/b331939.txt">Table of n, a(n) for n = 1..735</a> (the first 35 rows)
%H A331939 Wikipedia, <a href="https://en.wikipedia.org/wiki/Pentagon">Pentagon</a>.
%e A331939 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].
%e A331939 Triangle begins:
%e A331939   10,0,1,0
%e A331939   120,40,10,0,0
%e A331939   605,290,166,95,0,5
%e A331939   1750,1420,550,150,30,0,0
%e A331939   4315,3740,1920,640,95,20,5,6
%e A331939   9370,7950,3610,1200,220,20,10,0,0
%e A331939   17290,15705,7991,2885,520,75,20,5,0,0
%e A331939   29590,28130,13560,4320,860,150,0,0,0,0,0
%e A331939 The row sums are A331929.
%Y A331939 Cf A331929 (regions), A329710 (edges), A330847 (vertices), A331931, A331906, A007678, A092867, A331452.
%K A331939 nonn,tabf
%O A331939 1,1
%A A331939 _Scott R. Shannon_ and _N. J. A. Sloane_, Feb 02 2020