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A331451 Triangle read by rows: Take an n-sided polygon (n>=3) with all diagonals drawn, as in A007678. Then T(n,k) = number of k-sided polygons in that figure for k = 3, 4, ..., n.

%I A331451 #24 Mar 05 2020 22:49:57
%S A331451 1,4,0,10,0,1,18,6,0,0,35,7,7,0,1,56,24,0,0,0,0,90,36,18,9,0,0,1,120,
%T A331451 90,10,0,0,0,0,0,176,132,44,22,0,0,0,0,1,276,168,0,0,0,0,0,0,0,0,377,
%U A331451 234,117,39,0,13,0,0,0,0,1,476,378,98,0,0,0,0,0,0,0,0,0,585,600,150,105,15,0,0,0,0,0,0,0,1,848,672,128,48,0,0,0,0,0,0,0,0,0,0
%N A331451 Triangle read by rows: Take an n-sided polygon (n>=3) with all diagonals drawn, as in A007678. Then T(n,k) = number of k-sided polygons in that figure for k = 3, 4, ..., n.
%C A331451 Computed by _Scott R. Shannon_, Jan 24 2020
%H A331451 M. Rubinstein, <a href="/A006561/a006561_3.pdf">Drawings of A007678 for n=4,5,6,...</a>
%H A331451 Scott R. Shannon, <a href="/A331451/a331451.txt">Rows 3 through 45</a>
%H A331451 N. J. A. Sloane, <a href="/A331451/a331451.pdf">Illustration for row n=9</a>. [9-gon with one representative for each type of polygonal cell labeled with its number of sides]
%F A331451 By counting edges in two ways, we have the identity Sum_k k*T(n,k) + n = 2*A135565(n). E.g. for n=7, 3*35+4*7+5*7+6*0+7*1+7 = 182 = 2*A135565(7).
%e A331451 A hexagon with all diagonals drawn contains 18 triangles, 6 quadrilaterals, and no pentagons or hexagons, so row 6 is [18, 6, 0, 0].
%e A331451 Triangle begins:
%e A331451 1,
%e A331451 4,0,
%e A331451 10,0,1,
%e A331451 18,6,0,0,
%e A331451 35,7,7,0,1,
%e A331451 56,24,0,0,0,0,
%e A331451 90,36,18,9,0,0,1,
%e A331451 120,...
%e A331451 The row sums are A007678, the first column is A062361.
%Y A331451 Cf. A007678, A062361.
%Y A331451 See A331450 for a version of this triangle in which trailing zeros in the rows have been omitted.
%K A331451 nonn,tabl
%O A331451 3,2
%A A331451 _Scott R. Shannon_ and _N. J. A. Sloane_, Jan 25 2020