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A369524 Array read by antidiagonals: T(n,k) is the number of length n necklaces using at most k colors with black beads always occurring in runs of even length.

%I A369524 #27 Feb 19 2024 22:36:28
%S A369524 0,1,1,2,2,0,3,4,2,1,4,7,6,3,0,5,11,14,11,3,1,6,16,28,34,18,5,0,7,22,
%T A369524 50,87,81,38,5,1,8,29,82,191,276,227,70,8,0,9,37,126,373,759,983,615,
%U A369524 151,10,1,10,46,184,666,1782,3301,3500,1789,314,15,0,11,56,258,1109,3717,9180,14545,13007,5206,684,19,1
%N A369524 Array read by antidiagonals: T(n,k) is the number of length n necklaces using at most k colors with black beads always occurring in runs of even length.
%C A369524 Equivalently, black beads can be considered to have length 2, while all other beads have length 1.
%C A369524 Column k is the "CIK" (necklace, indistinct, unlabeled) transform of {k-1, 1, 0, 0, 0, ...} (see C. Bower link). - _Andrew Howroyd_, Jan 25 2024
%H A369524 C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>.
%F A369524 T(n,k) = (1/n) * Sum_{d|n} phi(n/d) * ((k-1)^d + Sum_{i=1..floor(d/2)} binomial(d-i-1,i-1) * d/i * (k-1)^(d-2*i)), where phi(n) = A000010.
%F A369524 G.f. of column k: Sum_{d>=1} (phi(d)/d) * log(1/(1 - (k-1)*x^d - x^(2*d))). - _Andrew Howroyd_, Jan 25 2024
%e A369524 n\k| 1  2   3     4      5       6       7        8         9 ...
%e A369524 ---+-----------------------------------------------------------------
%e A369524  1 | 0  1   2     3      4       5       6        7         8 ...A001477
%e A369524  2 | 1  2   4     7     11      16      22       29        37 ...A000124
%e A369524  3 | 0  2   6    14     28      50      82      126       184 ...A033547
%e A369524  4 | 1  3  11    34     87     191     373      666      1109
%e A369524  5 | 0  3  18    81    276     759    1782     3717      7080
%e A369524  6 | 1  5  38   227    983    3301    9180    22163     47997
%e A369524  7 | 0  5  70   615   3500   14545   48210   135155    333400
%e A369524  8 | 1  8 151  1789  13007   66166  260113   844691   2370229
%e A369524  9 | 0 10 314  5206  48820  304970 1423790  5358934  17110376
%e A369524 10 | 1 15 684 15490 186195 1425453 7897006 34438104 125093109
%e A369524 ...
%o A369524 (MATLAB)
%o A369524 function [res] = num2(n,k)
%o A369524 res=0;
%o A369524 for d=divisors(n)
%o A369524     s=(k-1)^d;
%o A369524     for i=1:floor(d/2)
%o A369524         s=s + nchoosek(d-i-1,i-1) * d/i * (k-1)^(d-2*i);
%o A369524     end
%o A369524     res= res + eulerPhi(n/d) * s;
%o A369524 end
%o A369524 res=res/n;
%o A369524 end
%o A369524 (PARI) T(n,k) = sum(d=1, n, eulerphi(d)*polcoef(log(1/(1 - (k-1)*x^d - x^(2*d)) + O(x*x^n)), n)/d)  \\ _Andrew Howroyd_, Jan 25 2024
%Y A369524 Columns 1..2 are A000035(n-1), A000358.
%Y A369524 Rows 1..3 are A001477(k-1), A000124(k-1), A033547(k-1).
%Y A369524 Cf. A000010 (phi), A075195 (all beads of same length).
%K A369524 nonn,tabl
%O A369524 1,4
%A A369524 _Maxim Karimov_ and Vladislav Sulima, Jan 25 2024