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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. 9
 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, 240, 120, 1, 46, 345, 900, 960, 360, 1, 62, 540, 1560, 1800, 720, 1, 94, 1173, 4980, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS The number of achiral necklaces is equivalent to the number of achiral bracelets. LINKS FORMULA 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. T(n,k) = 2*A273891(n,k) - A087854(n,k). 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 EXAMPLE The triangle begins with T(1,1): 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,    240,    120; 1,  46,   345,    900,    960,     360; 1,  62,   540,   1560,   1800,     720; 1,  94,  1173,   4980,   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; 1, 382, 11973, 113652, 480060, 1048320, 1234800,  745920, 181440; 1, 510, 18150, 186480, 834120, 1905120, 2328480, 1451520, 362880; For a(4,2)=4, the achiral loops are AAAB, AABB, ABAB, and ABBB. MATHEMATICA 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 PROG (PARI) T(n, k) = (k!/2)*(stirling(floor((n+1)/2), k, 2)+stirling(ceil((n+1)/2), k, 2)); tabf(nn) = for(n=1, nn, for (k=1, ceil((n+1)/2), print1(T(n, k), ", ")); print); \\ Michel Marcus, Jul 02 2018 CROSSREFS Odd rows are A019538. Even rows are A172106. Columns 1-6 are A057427, A027383, A056489, A056490, A056491, and A056492. Sequence in context: A344767 A187883 A134543 * A197871 A093010 A179000 Adjacent sequences:  A305537 A305538 A305539 * A305541 A305542 A305543 KEYWORD nonn,tabf,easy AUTHOR Robert A. Russell, Jun 04 2018 STATUS approved

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Last modified June 18 15:09 EDT 2021. Contains 345120 sequences. (Running on oeis4.)