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Triangle read by rows: T(n,k) is the number of achiral loops (necklaces or bracelets) of length n using exactly k different colors.
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%I #17 Sep 27 2018 12:34:35

%S 1,1,1,1,2,1,4,3,1,6,6,1,10,21,12,1,14,36,24,1,22,93,132,60,1,30,150,

%T 240,120,1,46,345,900,960,360,1,62,540,1560,1800,720,1,94,1173,4980,

%U 9300,7920,2520,1,126,1806,8400,16800,15120,5040,1,190,3801,24612,71400,103320,73080,20160,1,254,5796,40824,126000,191520,141120,40320

%N Triangle read by rows: T(n,k) is the number of achiral loops (necklaces or bracelets) of length n using exactly k different colors.

%C The number of achiral necklaces is equivalent to the number of achiral bracelets.

%F T(n,k) = (k!/2) * (S2(floor((n+1)/2),k) + S2(ceiling((n+1)/2),k)), where S2(n,k) is the Stirling subset number A008277.

%F T(n,k) = 2*A273891(n,k) - A087854(n,k).

%F G.f. for column k>1: (k!/2) * x^(2k-2) * (1+x)^2 / Product_{i=1..k} (1-i x^2). - _Robert A. Russell_, Sep 26 2018

%e The triangle begins with T(1,1):

%e 1;

%e 1, 1;

%e 1, 2;

%e 1, 4, 3;

%e 1, 6, 6;

%e 1, 10, 21, 12;

%e 1, 14, 36, 24;

%e 1, 22, 93, 132, 60;

%e 1, 30, 150, 240, 120;

%e 1, 46, 345, 900, 960, 360;

%e 1, 62, 540, 1560, 1800, 720;

%e 1, 94, 1173, 4980, 9300, 7920, 2520;

%e 1, 126, 1806, 8400, 16800, 15120, 5040;

%e 1, 190, 3801, 24612, 71400, 103320, 73080, 20160;

%e 1, 254, 5796, 40824, 126000, 191520, 141120, 40320;

%e 1, 382, 11973, 113652, 480060, 1048320, 1234800, 745920, 181440;

%e 1, 510, 18150, 186480, 834120, 1905120, 2328480, 1451520, 362880;

%e For a(4,2)=4, the achiral loops are AAAB, AABB, ABAB, and ABBB.

%t Table[(k!/2) (StirlingS2[Floor[(n + 1)/2], k] + StirlingS2[Ceiling[(n + 1)/2], k]), {n, 1, 15}, {k, 1, Ceiling[(n + 1)/2]}] // Flatten

%o (PARI) T(n, k) = (k!/2)*(stirling(floor((n+1)/2), k, 2)+stirling(ceil((n+1)/2), k, 2));

%o tabf(nn) = for(n=1, nn, for (k=1, ceil((n+1)/2), print1(T(n, k), ", ")); print); \\ _Michel Marcus_, Jul 02 2018

%Y Odd rows are A019538.

%Y Even rows are A172106.

%Y Columns 1-6 are A057427, A027383, A056489, A056490, A056491, and A056492.

%K nonn,tabf,easy

%O 1,5

%A _Robert A. Russell_, Jun 04 2018