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Numbers > 1 in A051168, with a(0) = 1.
10

%I #38 Aug 27 2024 03:01:47

%S 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 30,42,42,30,15,5,5,18,40,66,75,66,40,18,5,6,22,55,99,132,132,99,55,

%U 22,6,6,26,70,143,212,245,212,143,70,26,6,7,30,91,200,333,429,429,333,200

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

%C Original definition: Triangle read by rows in which row n gives the number of equal configurations under cyclic shift.

%C T(n,k) = A051168(n+3,k+1), if 0<k<=n. - _Michael Somos_, Jul 17 2004

%C From _Paul Weisenhorn_, Dec 21 2010 (Start):

%C 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 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 The numbers m=n*(n+3)/2 with 1<=n have one period with length (n+1).

%C The numbers m=n*(n+3)/2+2 with 1<=n have one period with length (n+2).

%C The triangular numbers n*(n+1)/2 with 1<=n have one period [(n+(n-1)+...+2+1)] with length 1. (End)

%H G. C. Greubel, <a href="/A092964/b092964.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%H 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 As triangle, starts:

%e 1;

%e 2,2;

%e 2,3,2;

%e 3,5,5,3;

%e 3,7,8,7,3;

%e 4,9,14,14,9,4;

%e 4,12,20,25,20,12,4; ...

%e From _Paul Weisenhorn_, Dec 21 2010: (Start)

%e 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 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 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 (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 Row sums give A093210. Essentially the same as A051168. See A185158 for another version.

%K nonn,tabl

%O 0,2

%A Thomas O. Hoffbauer (Thomas.Hoffbauer(AT)cibamberg.de), Apr 20 2004

%E Edited with better definition by _Omar E. Pol_, Jan 05 2009