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A331932 Triangle read by rows: Take a hexagon with all diagonals drawn, as in A331931. Then T(n,k) = number of k-sided polygons in that figure for k = 3, 4, ..., n+4.

%I A331932 #18 May 12 2020 06:10:44
%S A331932 18,6,0,264,108,36,0,1344,654,252,12,6,4164,2772,1020,228,24,0,10038,
%T A331932 7758,2424,516,72,24,0,21108,16188,6060,1128,156,0,0,0,39690,32022,
%U A331932 13368,3654,432,48,0,0,0,68052,56616,22980,6084,888,120,12,0,0,0
%N A331932 Triangle read by rows: Take a hexagon with all diagonals drawn, as in A331931. Then T(n,k) = number of k-sided polygons in that figure for k = 3, 4, ..., n+4.
%C A331932 See the links in A331931 for images of the hexagons.
%H A331932 Lars Blomberg, <a href="/A331932/b331932.txt">Table of n, a(n) for n = 1..525</a> (the first 30 rows)
%H A331932 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hexagon">Hexagon</a>.
%e A331932 A hexagon with no other points along its edges, n = 1, contains 18 triangles, 6 quadrilaterals and no other n-gons, so the first row is [18,6,0]. A hexagon with 1 point dividing its edges, n = 2, contains 264 triangles, 108 quadrilaterals, 36 pentagons and no other n-gons, so the second row is [264,108,36,0].
%e A331932 Triangle begins:
%e A331932   18,6,0
%e A331932   264,108,36,0
%e A331932   1344,654,252,12,6
%e A331932   4164,2772,1020,228,24,0
%e A331932   10038,7758,2424,516,72,24,0
%e A331932   21108,16188,6060,1128,156,0,0,0
%e A331932   39690,32022,13368,3654,432,48,0,0,0
%e A331932   68052,56616,22980,6084,888,120,12,0,0,0
%e A331932 The row sums are A331931.
%Y A331932 Cf. A331931 (regions), A330845 (edges), A330846 (vertices), A331906, A007678, A092867, A331452.
%K A331932 nonn,tabf
%O A331932 1,1
%A A331932 _Scott R. Shannon_ and _N. J. A. Sloane_, Feb 01 2020