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Triangle read by rows; T(n,k) is the number of achiral rows of n colors using exactly k colors.
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%I #11 Mar 21 2020 10:26:47

%S 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 30,150,240,120,0,1,30,150,240,120,0,1,62,540,1560,1800,720,0,1,62,

%U 540,1560,1800,720,0,1,126,1806,8400,16800,15120,5040,0,1,126,1806,8400,16800,15120,5040

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

%C Each zero in the data is the beginning of a new row.

%C Same as A131689, with rows (except for the first) repeated. - _Joerg Arndt_, Sep 08 2019

%F T(n,k) = k!*S2(ceiling(n/2),k), where S2 is the Stirling subset number A008277.

%e The triangle begins with T(0,0):

%e 1

%e 0 1

%e 0 1

%e 0 1 2

%e 0 1 2

%e 0 1 6 6

%e 0 1 6 6

%e 0 1 14 36 24

%e 0 1 14 36 24

%e 0 1 30 150 240 120

%e 0 1 30 150 240 120

%e 0 1 62 540 1560 1800 720

%e 0 1 62 540 1560 1800 720

%e 0 1 126 1806 8400 16800 15120 5040

%e 0 1 126 1806 8400 16800 15120 5040

%e 0 1 254 5796 40824 126000 191520 141120 40320

%e 0 1 254 5796 40824 126000 191520 141120 40320

%e 0 1 510 18150 186480 834120 1905120 2328480 1451520 362880

%e For T(7,2)=14, the rows are AAABAAA, AABABAA, AABBBAA, ABAAABA, ABABABA, ABBABBA, ABBBBBA, BAAAAAB, BAABAAB, BABABAB, BABBBAB, BBAAABB, BBABABB, and BBBABBB.

%t Table[k! StirlingS2[Ceiling[n/2], k], {n, 0, 18}, {k, 0, (n+1)/2}] // Flatten

%Y Columns 0-6 are A000007, A057427, A056453, A056454, A056455, A056456, A056457.

%Y Cf. A019538 (oriented), A305621 (unoriented), A305622 (chiral).

%Y Cf. A131689.

%K nonn,easy,tabf

%O 0,8

%A _Robert A. Russell_, Nov 09 2018