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%I #21 May 05 2021 00:19:01
%S 1,1,1,1,2,1,1,3,3,1,1,4,6,4,1,1,6,10,10,5,1,1,8,21,20,15,6,1,1,13,39,
%T 55,35,21,7,1,1,18,92,136,120,56,28,8,1,1,30,198,430,377,231,84,36,9,
%U 1,1,46,498,1300,1505,888,406,120,45,10,1
%N Table T(n,k) read by antidiagonals: number of necklaces allowing turnovers (bracelets) with n beads of k colors.
%C Unlike A075195 and A284855, antidiagonals go from bottom-left to top-right.
%H C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>
%F T(n, k) = (k^floor((n+1)/2) + k^ceiling((n+1)/2)) / 4 + (1/(2*n)) * Sum_{d divides n} phi(d) * k^(n/d). - _Robert A. Russell_, Sep 21 2018
%F G.f. for column k: (kx/4)*(kx+x+2)/(1-kx^2) - Sum_{d>0} phi(d)*log(1-kx^d)/2d. - _Robert A. Russell_, Sep 28 2018
%F T(n, k) = (k^floor((n+1)/2) + k^ceiling((n+1)/2))/4 + (1/(2*n))*Sum_{i=1..n} k^gcd(n,i). (See A075195 formulas.) - _Richard L. Ollerton_, May 04 2021
%e Table begins with T[0,1]:
%e 1 1 1 1 1 1 1 1 1 1
%e 1 2 3 4 5 6 7 8 9 10
%e 1 3 6 10 15 21 28 36 45 55
%e 1 4 10 20 35 56 84 120 165 220
%e 1 6 21 55 120 231 406 666 1035 1540
%e 1 8 39 136 377 888 1855 3536 6273 10504
%e 1 13 92 430 1505 4291 10528 23052 46185 86185
%e 1 18 198 1300 5895 20646 60028 151848 344925 719290
%e 1 30 498 4435 25395 107331 365260 1058058 2707245 6278140
%e 1 46 1219 15084 110085 563786 2250311 7472984 21552969 55605670
%e 1 78 3210 53764 493131 3037314 14158228 53762472 174489813 500280022
%t b[n_, k_] := DivisorSum[n, EulerPhi[#]*k^(n/#) &] / n;
%t c[n_, k_] := If[EvenQ[n], (k^(n/2) + k^(n/2+1))/2, k^((n+1)/2)];
%t T[0, _] = 1; T[n_, k_] := (b[n, k] + c[n, k])/2;
%t Table[T[n, k-n], {k, 1, 11}, {n, k-1, 0, -1}] // Flatten
%t {* _Robert A. Russell_, Sep 21 2018 after Jean-François Alcover *)
%Y Columns 2-6 are A000029, A027671, A032275, A032276, and A056341.
%Y Rows 2-7 are A000217, A000292, A002817, A060446, A027670, and A060532.
%Y Cf. A000031.
%Y Cf. A081720, A081721.
%Y T(n,k) = (A075195(n,k) + A284855(n,k)) / 2.
%K nice,nonn,tabl
%O 0,5
%A _Alford Arnold_
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