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Triangle read by rows: CU(n,k) is the number of unlabeled subgraphs with k edges of the n-cycle C_n.
3

%I #24 Sep 27 2019 15:22:48

%S 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,

%T 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,

%U 21,13,6,1,1,11,10,17,21,28,29,31,24,16,6,1,1

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

%H Andrew Howroyd, <a href="/A278427/b278427.txt">Table of n, a(n) for n = 0..1274</a>

%F T(n,n) = 1; T(n,k) = Sum_{i=k+1..n} A008284(i, i-k) for k < n. - _Andrew Howroyd_, Sep 26 2019

%e 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).

%e Triangle begins:

%e 1;

%e 1, 1;

%e 2, 1, 1;

%e 3, 2, 1, 1;

%e 4, 3, 3, 1, 1;

%e 5, 4, 5, 3, 1, 1;

%e 6, 5, 7, 6, 4, 1, 1;

%e 7, 6, 9, 9, 8, 4, 1, 1;

%e 8, 7, 11, 12, 13, 9, 5, 1, 1;

%e 9, 8, 13, 15, 18, 15, 12, 5, 1, 1;

%e 10, 9, 15, 18, 23, 22, 21, 13, 6, 1, 1;

%e ...

%o (PARI) \\ here P is A008284 as vector of vectors.

%o P(n)={[Vecrev(p/y) | p<-Vec(-1 + 1/prod(k=1, n, 1 - y*x^k + O(x*x^n)))]}

%o 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])))}

%o { my(A=T(12)); for(n=1, #A, print(A[n,1..n])) } \\ _Andrew Howroyd_, Sep 26 2019

%Y Cf. A008284.

%Y Rows sums give A000070.

%Y Middle diagonal gives A058397.

%K nonn,tabl

%O 0,4

%A _John P. McSorley_, Nov 21 2016

%E Offset corrected and terms a(66) and beyond from _Andrew Howroyd_, Sep 26 2019