cp's OEIS Frontend

A051137 Table T(n,k) read by antidiagonals: number of necklaces allowing turnovers (bracelets) with n beads of k colors.

%I A051137 #26 Aug 12 2024 01:27:48
%S A051137 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 A051137 55,35,21,7,1,1,18,92,136,120,56,28,8,1,1,30,198,430,377,231,84,36,9,
%U A051137 1,1,46,498,1300,1505,888,406,120,45,10,1
%N A051137 Table T(n,k) read by antidiagonals: number of necklaces allowing turnovers (bracelets) with n beads of k colors.
%C A051137 Unlike A075195 and A284855, antidiagonals go from bottom-left to top-right.
%H A051137 C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>
%F A051137 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 A051137 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 A051137 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 A051137 Table begins with T[0,1]:
%e A051137 1  1    1     1      1       1        1        1         1         1
%e A051137 1  2    3     4      5       6        7        8         9        10
%e A051137 1  3    6    10     15      21       28       36        45        55
%e A051137 1  4   10    20     35      56       84      120       165       220
%e A051137 1  6   21    55    120     231      406      666      1035      1540
%e A051137 1  8   39   136    377     888     1855     3536      6273     10504
%e A051137 1 13   92   430   1505    4291    10528    23052     46185     86185
%e A051137 1 18  198  1300   5895   20646    60028   151848    344925    719290
%e A051137 1 30  498  4435  25395  107331   365260  1058058   2707245   6278140
%e A051137 1 46 1219 15084 110085  563786  2250311  7472984  21552969  55605670
%e A051137 1 78 3210 53764 493131 3037314 14158228 53762472 174489813 500280022
%t A051137 b[n_, k_] := DivisorSum[n, EulerPhi[#]*k^(n/#) &] / n;
%t A051137 c[n_, k_] := If[EvenQ[n], (k^(n/2) + k^(n/2+1))/2, k^((n+1)/2)];
%t A051137 T[0, _] = 1; T[n_, k_] := (b[n, k] + c[n, k])/2;
%t A051137 Table[T[n, k-n], {k, 1, 11}, {n, k-1, 0, -1}] // Flatten
%t A051137 (* _Robert A. Russell_, Sep 21 2018 after _Jean-François Alcover_ *)
%Y A051137 Columns 2-6 are A000029, A027671, A032275, A032276, and A056341.
%Y A051137 Rows 2-7 are A000217, A000292, A002817, A060446, A027670, and A060532.
%Y A051137 Cf. A000031.
%Y A051137 Cf. A081720, A081721.
%Y A051137 T(n,k) = (A075195(n,k) + A284855(n,k)) / 2.
%K A051137 nice,nonn,tabl
%O A051137 0,5
%A A051137 _Alford Arnold_