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.

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

Offset: 1

Author

Robert A. Russell, Jun 04 2018

Keywords

• nonn
• tabf
• easy

Comments

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

Examples

			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.
		

Crossrefs

Odd rows are A019538.

Even rows are A172106.

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

Programs

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

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

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