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A305540 Triangle read by rows: T(n,k) is the number of achiral loops (necklaces or bracelets) of length n using exactly k different colors.

%I A305540 #17 Sep 27 2018 12:34:35
%S A305540 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 A305540 240,120,1,46,345,900,960,360,1,62,540,1560,1800,720,1,94,1173,4980,
%U A305540 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 A305540 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 A305540 The number of achiral necklaces is equivalent to the number of achiral bracelets.
%F A305540 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 A305540 T(n,k) = 2*A273891(n,k) - A087854(n,k).
%F A305540 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 A305540 The triangle begins with T(1,1):
%e A305540 1;
%e A305540 1,   1;
%e A305540 1,   2;
%e A305540 1,   4,     3;
%e A305540 1,   6,     6;
%e A305540 1,  10,    21,     12;
%e A305540 1,  14,    36,     24;
%e A305540 1,  22,    93,    132,     60;
%e A305540 1,  30,   150,    240,    120;
%e A305540 1,  46,   345,    900,    960,     360;
%e A305540 1,  62,   540,   1560,   1800,     720;
%e A305540 1,  94,  1173,   4980,   9300,    7920,    2520;
%e A305540 1, 126,  1806,   8400,  16800,   15120,    5040;
%e A305540 1, 190,  3801,  24612,  71400,  103320,   73080,   20160;
%e A305540 1, 254,  5796,  40824, 126000,  191520,  141120,   40320;
%e A305540 1, 382, 11973, 113652, 480060, 1048320, 1234800,  745920, 181440;
%e A305540 1, 510, 18150, 186480, 834120, 1905120, 2328480, 1451520, 362880;
%e A305540 For a(4,2)=4, the achiral loops are AAAB, AABB, ABAB, and ABBB.
%t A305540 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 A305540 (PARI) T(n, k) = (k!/2)*(stirling(floor((n+1)/2), k, 2)+stirling(ceil((n+1)/2), k, 2));
%o A305540 tabf(nn) = for(n=1, nn, for (k=1, ceil((n+1)/2), print1(T(n, k), ", ")); print); \\ _Michel Marcus_, Jul 02 2018
%Y A305540 Odd rows are A019538.
%Y A305540 Even rows are A172106.
%Y A305540 Columns 1-6 are A057427, A027383, A056489, A056490, A056491, and A056492.
%K A305540 nonn,tabf,easy
%O A305540 1,5
%A A305540 _Robert A. Russell_, Jun 04 2018