A278427 Triangle read by rows: CU(n,k) is the number of unlabeled subgraphs with k edges of the n-cycle C_n.

1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 3, 3, 1, 1, 5, 4, 5, 3, 1, 1, 6, 5, 7, 6, 4, 1, 1, 7, 6, 9, 9, 8, 4, 1, 1, 8, 7, 11, 12, 13, 9, 5, 1, 1, 9, 8, 13, 15, 18, 15, 12, 5, 1, 1, 10, 9, 15, 18, 23, 22, 21, 13, 6, 1, 1, 11, 10, 17, 21, 28, 29, 31, 24, 16, 6, 1, 1

Offset: 0

Author

John P. McSorley, Nov 21 2016

Keywords

• nonn
• tabl

Examples

			For row n = 3 of the triangle below: there are 3 unlabeled subgraphs of the triangle C_3 with 0 edges, 2 with 1 edge, 1 with 2 edges, and 1 with 3 edges (C_3 itself).
Triangle begins:
   1;
   1,  1;
   2,  1,  1;
   3,  2,  1,  1;
   4,  3,  3,  1,  1;
   5,  4,  5,  3,  1,  1;
   6,  5,  7,  6,  4,  1,  1;
   7,  6,  9,  9,  8,  4,  1,  1;
   8,  7, 11, 12, 13,  9,  5,  1,  1;
   9,  8, 13, 15, 18, 15, 12,  5,  1,  1;
  10,  9, 15, 18, 23, 22, 21, 13,  6,  1,  1;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1274

Crossrefs

Cf. A008284.

Rows sums give A000070.

Middle diagonal gives A058397.

Programs

PARI

\\ here P is A008284 as vector of vectors.
P(n)={[Vecrev(p/y) | p<-Vec(-1 + 1/prod(k=1, n, 1 - y*x^k + O(x*x^n)))]}
T(n)={my(p=P(n-1)); matrix(n, n, n, k, if(k>=n, k==n, sum(i=k, n-1, p[i][i-k+1])))}
{ my(A=T(12)); for(n=1, #A, print(A[n,1..n])) } \\ Andrew Howroyd, Sep 26 2019

Formula

T(n,n) = 1; T(n,k) = Sum_{i=k+1..n} A008284(i, i-k) for k < n. - Andrew Howroyd, Sep 26 2019

Extensions

Offset corrected and terms a(66) and beyond from Andrew Howroyd, Sep 26 2019