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

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, 30, 42, 42, 30, 15, 5, 5, 18, 40, 66, 75, 66, 40, 18, 5, 6, 22, 55, 99, 132, 132, 99, 55, 22, 6, 6, 26, 70, 143, 212, 245, 212, 143, 70, 26, 6, 7, 30, 91, 200, 333, 429, 429, 333, 200

Offset: 0

Author

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

Keywords

• nonn
• tabl

Comments

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

T(n,k) = A051168(n+3,k+1), if 0Michael Somos, Jul 17 2004

From Paul Weisenhorn, Dec 21 2010 (Start):

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.

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).

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

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

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

Examples

			As triangle, starts:
  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; ...
From _Paul Weisenhorn_, Dec 21 2010: (Start)
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)].
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)
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
• Pieter Moree, The formal series Witt transform, Discr. Math. no. 295 vol. 1-3 (2005) 143-160.

Crossrefs

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

Programs

Mathematica

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 *)

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 */

Extensions

Edited with better definition by Omar E. Pol, Jan 05 2009