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A283615 Irregular triangle read by rows: T(n,k) is the number of necklaces of n 1's, n -1's, and k 0's such that no two adjacent elements are equal.

%I A283615 #21 Aug 02 2022 09:12:55
%S A283615 1,1,2,1,1,2,5,4,2,1,2,7,16,18,12,4,1,2,11,32,70,92,82,40,10,1,2,13,
%T A283615 56,166,348,510,520,350,140,26,1,2,17,88,336,932,1948,2992,3404,2756,
%U A283615 1518,504,80,1,2,19,124,584,2056,5524,11444,18298,22428,20706,13944,6468,1848,246,1,2,23,168,944,3976,13120,34064,70380,115516
%N A283615 Irregular triangle read by rows: T(n,k) is the number of necklaces of n 1's, n -1's, and k 0's such that no two adjacent elements are equal.
%C A283615 T(n,k) is the number of unique circular arrays (A283614) given equivalence under rotation.
%F A283615 T(n,k) = Sum_{d|gcd(n,k)} phi(d) * A283614(n/d,k/d) / (2*n+k) where phi is Euler's totient function (A000010).
%F A283615 T(n,2*n) = A003239(n).
%F A283615 T(n,2*n-1) = 2*binomial(2*(n-1), n-1).
%F A283615 T(n,n) = A110710(n).
%e A283615 Table for n=[0..6], k=[0..12]
%e A283615     0 1  2   3    4     5     6      7      8       9      10      11      12
%e A283615 -----------------------------------------------------------------------------
%e A283615 0 | 1
%e A283615 1 | 1 2  1
%e A283615 2 | 1 2  5   4    2
%e A283615 3 | 1 2  7  16   18    12     4
%e A283615 4 | 1 2 11  32   70    92    82     40     10
%e A283615 5 | 1 2 13  56  166   348   510    520    350     140      26
%e A283615 6 | 1 2 17  88  336   932  1948   2992   3404    2756    1518     504      80
%e A283615 The 13 necklaces for n=5, k=2 are:
%e A283615 [+-+-+-+-0+0-],[+-+-+-+0+-0-],[+-+-+-+0-+0-],[+-+-+-0+-+0-]
%e A283615 [+-+-+0+-+-0-],[+-+-+0-+-+0-],[+-+-+-+-+0-0],[+-+-+-+-0+-0]
%e A283615 [+-+-+-+-0-+0],[+-+-+-+0-+-0],[+-+-+-0+-+-0],[+-+-+-0-+-+0]
%e A283615 [+-+-+0-+-+-0].
%o A283615 (Maxima)
%o A283615 g(x,y):=2*(x*y+1)/sqrt((1-y)*(1-(2*x+1)^2*y))-1;
%o A283615 A283614(n,k):=coeff(limit(diff(g(x,y),y,n)/n!,y,0),x,k);
%o A283615 A283615(n,k):=block([s,d],
%o A283615   s:0,
%o A283615   for d in divisors(gcd(n,k)) do
%o A283615     s:s+totient(d)*A283614(n/d,k/d),
%o A283615   return(s/(2*n+k)));
%Y A283615 Cf. A000010, A003239, A110710, A283614.
%K A283615 nonn,tabf
%O A283615 0,3
%A A283615 _Stefan Hollos_, Apr 11 2017