login
Triangular array: a(n,k) is the number of binary necklaces (no turning over) of length n whose longest run of 1's has length k. Table begins at n=0, k=0.
1

%I #7 Oct 15 2017 08:41:07

%S 1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,2,2,1,1,1,1,4,4,2,1,1,1,1,4,6,4,2,1,

%T 1,1,1,7,11,8,4,2,1,1,1,1,9,19,14,8,4,2,1,1,1,1,14,33,27,16,8,4,2,1,1,

%U 1,1,18,56,50,30,16,8,4,2,1,1,1,1,30,101,96,59,32,16,8,4,2,1,1,1

%N Triangular array: a(n,k) is the number of binary necklaces (no turning over) of length n whose longest run of 1's has length k. Table begins at n=0, k=0.

%C Column k=1 appears to be A032190(n), n=2,3,...

%H Andrew Howroyd, <a href="/A063686/b063686.txt">Table of n, a(n) for n = 0..1274</a>

%e Triangle begins:

%e 1;

%e 1, 1;

%e 1, 1, 1;

%e 1, 1, 1, 1;

%e 1, 2, 1, 1, 1;

%e 1, 2, 2, 1, 1, 1;

%e 1, 4, 4, 2, 1, 1, 1;

%e 1, 4, 6, 4, 2, 1, 1, 1;

%e 1, 7, 11, 8, 4, 2, 1, 1, 1;

%e 1, 9, 19, 14, 8, 4, 2, 1, 1, 1;

%e 1, 14, 33, 27, 16, 8, 4, 2, 1, 1, 1;

%e ...

%o (PARI) \\ here R(n) is A048887 transposed

%o R(n)={Mat(vector(n, k, Col((1-x)/(1-2*x+x^(k+1)) - 1 + O(x*x^n))))}

%o S(M)={matrix(#M-1, #M-1, n, k, if(k<n-1, (k+1)*M[n-k-1,k+1] + sum(j=1, k, j*(M[n-j,k+1]-M[n-j,k])), if(k<n, n)))}

%o T(n)={my(M=S(R(n+1))); matid(n) + matrix(n, n, n, k, sumdiv(n, d, if(k<d, eulerphi(n/d)*M[d, k]))/n) }

%o my(M=T(10)); for(n=0, #M, for(k=0, n, print1(if(k==0,1,M[n,k]), ", ")); print) \\ _Andrew Howroyd_, Oct 15 2017

%Y Cf. A032190, A048004, A048887.

%Y Cf. A000358, A093305, A280218 (necklaces avoiding 00, 000, 0000).

%K nonn,tabl

%O 0,12

%A Christopher Lenard (c.lenard(AT)bendigo.latrobe.edu.au), Aug 22 2001

%E a(0,0)=1 from _Andrew Howroyd_, Oct 15 2017