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 A305541 Triangle read by rows: T(n,k) is the number of chiral pairs of color loops of length n with exactly k different colors. 9
 0, 0, 0, 0, 0, 1, 0, 0, 3, 3, 0, 0, 12, 24, 12, 0, 1, 35, 124, 150, 60, 0, 2, 111, 588, 1200, 1080, 360, 0, 6, 318, 2487, 7845, 11970, 8820, 2520, 0, 14, 934, 10240, 46280, 105840, 129360, 80640, 20160, 0, 30, 2634, 40488, 254676, 821592, 1481760, 1512000, 816480, 181440, 0, 62, 7503, 158220, 1344900, 5873760, 14658840, 21772800, 19051200, 9072000, 1814400 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,9 COMMENTS In other words, the number of n-bead bracelets with beads of exactly k different colors that when turned over are different from themselves. - Andrew Howroyd, Sep 13 2019 LINKS Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows) FORMULA T(n,k) = -(k!/4)*(S2(floor((n+1)/2),k) + S2(ceiling((n+1)/2),k)) + (k!/(2 n))*Sum_{d|n} phi(d)*S2(n/d,k), where S2(n,k) is the Stirling subset number A008277. T(n,k) = A087854(n,k) - A273891(n,k). T(n,k) = (A087854(n,k) - A305540(n,k)) / 2. T(n, k) = Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*A293496(n, i). - Andrew Howroyd, Sep 13 2019 EXAMPLE Triangle T(n,k) begins:   0;   0,  0;   0,  0,    1;   0,  0,    3,     3;   0,  0,   12,    24,     12;   0,  1,   35,   124,    150,     60;   0,  2,  111,   588,   1200,   1080,     360;   0,  6,  318,  2487,   7845,  11970,    8820,    2520;   0, 14,  934, 10240,  46280, 105840,  129360,   80640,  20160;   0, 30, 2634, 40488, 254676, 821592, 1481760, 1512000, 816480, 181440;   ... For T(4,3)=3, the chiral pairs are AABC-AACB, ABBC-ACBB, and ABCC-ACCB. For T(4,4)=3, the chiral pairs are ABCD-ADCB, ABDC-ACDB, and ACBD-ADBC. MATHEMATICA Table[(k!/(2n)) DivisorSum[n, EulerPhi[#] StirlingS2[n/#, k] &] - (k!/4) (StirlingS2[Floor[(n+1)/2], k] + StirlingS2[Ceiling[(n+1)/2], k]), {n, 1, 15}, {k, 1, n}] // Flatten PROG (PARI) T(n, k) = {-k!*(stirling((n+1)\2, k, 2) + stirling(n\2+1, k, 2))/4 + k!*sumdiv(n, d, eulerphi(d)*stirling(n/d, k, 2))/(2*n)} \\ Andrew Howroyd, Sep 13 2019 CROSSREFS Columns 2-6 are A059076, A305542, A305543, A305544, and A305545. Row sums are A326895. Cf. A087854, A273891, A293496, A305540, A309651. Sequence in context: A199041 A199237 A309651 * A280810 A283386 A278385 Adjacent sequences:  A305538 A305539 A305540 * A305542 A305543 A305544 KEYWORD nonn,tabl,easy AUTHOR Robert A. Russell, Jun 04 2018 STATUS approved

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Last modified January 16 14:39 EST 2022. Contains 350376 sequences. (Running on oeis4.)