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A321434 Triangle read by rows; T(n,k) is the number of achiral rows of n colors using exactly k colors.

%I A321434 #11 Mar 21 2020 10:26:47
%S A321434 1,0,1,0,1,0,1,2,0,1,2,0,1,6,6,0,1,6,6,0,1,14,36,24,0,1,14,36,24,0,1,
%T A321434 30,150,240,120,0,1,30,150,240,120,0,1,62,540,1560,1800,720,0,1,62,
%U A321434 540,1560,1800,720,0,1,126,1806,8400,16800,15120,5040,0,1,126,1806,8400,16800,15120,5040
%N A321434 Triangle read by rows; T(n,k) is the number of achiral rows of n colors using exactly k colors.
%C A321434 Each zero in the data is the beginning of a new row.
%C A321434 Same as A131689, with rows (except for the first) repeated. - _Joerg Arndt_, Sep 08 2019
%F A321434 T(n,k) = k!*S2(ceiling(n/2),k), where S2 is the Stirling subset number A008277.
%e A321434 The triangle begins with T(0,0):
%e A321434 1
%e A321434 0 1
%e A321434 0 1
%e A321434 0 1   2
%e A321434 0 1   2
%e A321434 0 1   6     6
%e A321434 0 1   6     6
%e A321434 0 1  14    36     24
%e A321434 0 1  14    36     24
%e A321434 0 1  30   150    240    120
%e A321434 0 1  30   150    240    120
%e A321434 0 1  62   540   1560   1800    720
%e A321434 0 1  62   540   1560   1800    720
%e A321434 0 1 126  1806   8400  16800   15120    5040
%e A321434 0 1 126  1806   8400  16800   15120    5040
%e A321434 0 1 254  5796  40824 126000  191520  141120   40320
%e A321434 0 1 254  5796  40824 126000  191520  141120   40320
%e A321434 0 1 510 18150 186480 834120 1905120 2328480 1451520 362880
%e A321434 For T(7,2)=14, the rows are AAABAAA, AABABAA, AABBBAA, ABAAABA, ABABABA, ABBABBA, ABBBBBA, BAAAAAB, BAABAAB, BABABAB, BABBBAB, BBAAABB, BBABABB, and BBBABBB.
%t A321434 Table[k! StirlingS2[Ceiling[n/2], k], {n, 0, 18}, {k, 0, (n+1)/2}] // Flatten
%Y A321434 Columns 0-6 are A000007, A057427, A056453, A056454, A056455, A056456, A056457.
%Y A321434 Cf. A019538 (oriented), A305621 (unoriented), A305622 (chiral).
%Y A321434 Cf. A131689.
%K A321434 nonn,easy,tabf
%O A321434 0,8
%A A321434 _Robert A. Russell_, Nov 09 2018