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A092964 Numbers > 1 in A051168, with a(0) = 1.

%I A092964 #38 Aug 27 2024 03:01:47
%S A092964 1,2,2,2,3,2,3,5,5,3,3,7,8,7,3,4,9,14,14,9,4,4,12,20,25,20,12,4,5,15,
%T A092964 30,42,42,30,15,5,5,18,40,66,75,66,40,18,5,6,22,55,99,132,132,99,55,
%U A092964 22,6,6,26,70,143,212,245,212,143,70,26,6,7,30,91,200,333,429,429,333,200
%N A092964 Numbers > 1 in A051168, with a(0) = 1.
%C A092964 Original definition: Triangle read by rows in which row n gives the number of equal configurations under cyclic shift.
%C A092964 T(n,k) = A051168(n+3,k+1), if 0<k<=n. - _Michael Somos_, Jul 17 2004
%C A092964 From _Paul Weisenhorn_, Dec 21 2010 (Start):
%C A092964 T(n,k) is the number of classes that have (k+1) ordered sums of (n+4) with (k+1) positive integers, that can be transformed into each other by a cyclic permutation.
%C A092964 m has 2^(m-1) ordered sums; for each sum one remove the first part z(1) and add 1 to the next z(1) parts to get a new ordered sum until a period is reached. T(n,k)=a(m) with m=(n+4)*(n+3)/2+k+1 gives for m the number of periods with length (n+4).
%C A092964 The numbers m=n*(n+3)/2 with 1<=n have one period with length (n+1).
%C A092964 The numbers m=n*(n+3)/2+2 with 1<=n have one period with length (n+2).
%C A092964 The triangular numbers n*(n+1)/2 with 1<=n have one period [(n+(n-1)+...+2+1)] with length 1. (End)
%H A092964 G. C. Greubel, <a href="/A092964/b092964.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A092964 Pieter Moree, <a href="http://dx.doi.org/10.1016/j.disc.2005.03.004">The formal series Witt transform</a>, Discr. Math. no. 295 vol. 1-3 (2005) 143-160.
%e A092964 As triangle, starts:
%e A092964   1;
%e A092964   2,2;
%e A092964   2,3,2;
%e A092964   3,5,5,3;
%e A092964   3,7,8,7,3;
%e A092964   4,9,14,14,9,4;
%e A092964   4,12,20,25,20,12,4; ...
%e A092964 From _Paul Weisenhorn_, Dec 21 2010: (Start)
%e A092964 T(2,2)=3 classes with 3 ordered sums of 6; [(1+1+4),(1+4+1),(4+1+1)]; [(1+2+3),(2+3+1),(3+1+2)]; [(1+3+2),(3+2+1),(2+1+3)].
%e A092964 T(2,2)=a(m)=3 periods with length 6 for m=6*5/2+3=18 [(5+5+4+3+1),(6+5+4+2+1),(6+5+3+2+1+1),(6+4+3+2+2+1),(5+4+3+3+2+1),(5+4+4+3+2)]; [(5+5+3+3+2),(6+4+4+3+1),(5+5+4+2+1+1),(6+5+3+2+2),(6+4+3+3+1+1),(5+4+4+2+2+1)]; [(5+5+3+2+2+1),(6+4+3+3+2),(5+4+4+3+1+1),(5+5+4+2+2),(6+5+3+3+1),(6+4+4+2+1+1)]. (End)
%t A092964 T[n_, k_] := DivisorSum[GCD[k + 1, n + 4], Binomial[(n + 4)/#, (k + 1)/#] * MoebiusMu[#] & ]/(n + 4); Table[T[n, k], {n, 0, 12}, {k, 1, n + 1}] // Flatten (* _Jean-François Alcover_, Dec 02 2015 *)
%o A092964 (PARI) T(n,k)=local(A,ps,c); n+=3; k++; if(k<1||k>=n-1, 0, A=x*O(x^n) + y*O(y^n); ps=1-x-y+A; for(m=1,n,for(i=0,m,c=polcoeff(polcoeff(ps,i,x),m-i, y); if(m==n&i==k,break(2),ps*=(1-y^(m-i)*x^i+A)^c)));-c) /* _Michael Somos_, Jul 17 2004 */
%Y A092964 Row sums give A093210. Essentially the same as A051168. See A185158 for another version.
%K A092964 nonn,tabl
%O A092964 0,2
%A A092964 Thomas O. Hoffbauer (Thomas.Hoffbauer(AT)cibamberg.de), Apr 20 2004
%E A092964 Edited with better definition by _Omar E. Pol_, Jan 05 2009