cp's OEIS Frontend

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.

%I A305541 #14 Sep 14 2019 12:20:10
%S A305541 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,
%T A305541 1080,360,0,6,318,2487,7845,11970,8820,2520,0,14,934,10240,46280,
%U A305541 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
%N 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.
%C A305541 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
%H A305541 Andrew Howroyd, <a href="/A305541/b305541.txt">Table of n, a(n) for n = 1..1275</a> (first 50 rows)
%F A305541 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.
%F A305541 T(n,k) = A087854(n,k) - A273891(n,k).
%F A305541 T(n,k) = (A087854(n,k) - A305540(n,k)) / 2.
%F A305541 T(n, k) = Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*A293496(n, i). - _Andrew Howroyd_, Sep 13 2019
%e A305541 Triangle T(n,k) begins:
%e A305541   0;
%e A305541   0,  0;
%e A305541   0,  0,    1;
%e A305541   0,  0,    3,     3;
%e A305541   0,  0,   12,    24,     12;
%e A305541   0,  1,   35,   124,    150,     60;
%e A305541   0,  2,  111,   588,   1200,   1080,     360;
%e A305541   0,  6,  318,  2487,   7845,  11970,    8820,    2520;
%e A305541   0, 14,  934, 10240,  46280, 105840,  129360,   80640,  20160;
%e A305541   0, 30, 2634, 40488, 254676, 821592, 1481760, 1512000, 816480, 181440;
%e A305541   ...
%e A305541 For T(4,3)=3, the chiral pairs are AABC-AACB, ABBC-ACBB, and ABCC-ACCB.
%e A305541 For T(4,4)=3, the chiral pairs are ABCD-ADCB, ABDC-ACDB, and ACBD-ADBC.
%t A305541 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
%o A305541 (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
%Y A305541 Columns 2-6 are A059076, A305542, A305543, A305544, and A305545.
%Y A305541 Row sums are A326895.
%Y A305541 Cf. A087854, A273891, A293496, A305540, A309651.
%K A305541 nonn,tabl,easy
%O A305541 1,9
%A A305541 _Robert A. Russell_, Jun 04 2018