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%I #21 Apr 13 2013 08:47:33
%S 0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,2,1,0,0,1,0,0,0,0,0,5,2,
%T 1,0,0,1,0,2,0,0,0,2,21,12,1,1,0,0,2,0,31,0,0,0,0,3,103,220,7,1,1,0,0,
%U 3,0,1606,0,1,0,0,0,5,752,16829,388,9,1,1,0,0,5,0,193900,0,6,0,0,0
%N Triangular array E(n,k) counting, not necessarily connected, k-regular simple graphs on n vertices with girth exactly 4.
%C In the n-th row 0 <= 2k <= n.
%H Jason Kimberley, <a href="/A185644/b185644.txt">Table of n, a(i)=E(n,k) for i = 1..139 (n = 1..22)</a>
%H Jason Kimberley, <a href="/wiki/User:Jason_Kimberley/E_k-reg_girth_eq_g_index">Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g</a>
%F E(n,k) = A186734(n,k) + A210704(n,k), noting the differing row lengths.
%F E(n,k) = A185304(n,k) - A185305(n,k), noting the differing row lengths.
%e 01: 0;
%e 02: 0, 0;
%e 03: 0, 0;
%e 04: 0, 0, 1;
%e 05: 0, 0, 0;
%e 06: 0, 0, 0, 1;
%e 07: 0, 0, 0, 0;
%e 08: 0, 0, 1, 2, 1;
%e 09: 0, 0, 1, 0, 0;
%e 10: 0, 0, 0, 5, 2, 1;
%e 11: 0, 0, 1, 0, 2, 0;
%e 12: 0, 0, 2, 21, 12, 1, 1;
%e 13: 0, 0, 2, 0, 31, 0, 0;
%e 14: 0, 0, 3, 103, 220, 7, 1, 1;
%e 15: 0, 0, 3, 0, 1606, 0, 1, 0;
%e 16: 0, 0, 5, 752, 16829, 388, 9, 1, 1;
%e 17: 0, 0, 5, 0, 193900, 0, 6, 0, 0;
%e 18: 0, 0, 7, 7385, 2452820, 406824, 267, 8, 1, 1;
%e 19: 0, 0, 8, 0, 32670331, 0, 3727, 0, 0, 0;
%e 20: 0, 0, 11, 91939, 456028487, 1125022326, 483012, 741, 13, 1, 1;
%e 21: 0, 0, 12, 0, 6636066126, 0, 69823723, 0, 1, 0, 0;
%e 22: 0, 0, 16, 1345933, 100135577863, 3813549359275, 14836130862, 2887493, ?, 14, 1;
%Y The sum of the n-th row of this sequence is A198314(n).
%Y Not necessarily connected k-regular simple graphs girth exactly 4: A198314 (any k), this sequence (triangle); fixed k: A026797 (k=2), A185134 (k=3), A185144 (k=4).
%K nonn,hard,tabf
%O 1,23
%A _Jason Kimberley_, Feb 22 2013
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