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%I #45 May 20 2020 02:43:37
%S 1,0,1,0,0,1,0,0,1,1,0,0,1,0,1,0,0,1,2,1,1,0,0,1,0,2,0,1,0,0,1,5,6,3,
%T 1,1,0,0,1,0,16,0,4,0,1,0,0,1,19,59,60,21,5,1,1,0,0,1,0,265,0,266,0,6,
%U 0,1,0,0,1,85,1544,7848,7849,1547,94,9,1,1,0,0,1,0,10778,0,367860,0
%N Triangular array C(n, r) = number of connected r-regular graphs with n nodes, 0 <= r < n.
%C A graph is called r-regular if every node has exactly r edges. The numbers in this table were copied from the column sequences.
%C This sequence can be derived from A051031 by inverse Euler transform. See the comments in A051031 for a brief description of how that sequence can be computed without generating all regular graphs. - _Andrew Howroyd_, Mar 13 2020
%H Andrew Howroyd, <a href="/A068934/b068934.txt">Table of n, a(n) for n = 1..300</a> (rows 1..24, first 16 rows from Jason Kimberley)
%H Jason Kimberley, <a href="/wiki/User:Jason_Kimberley/A068934">Connected regular graphs (with girth at least 3)</a>
%H Jason Kimberley, <a href="/wiki/User:Jason_Kimberley/C_k-reg_girth_ge_g_index">Index of sequences counting connected k-regular simple graphs with girth at least g</a>
%H Zhipeng Xu, Xiaolong Huang, Fabian Jimenez, Yuefan Deng, <a href="https://arxiv.org/abs/1907.12455">A new record of enumeration of regular graphs by parallel processing</a>, arXiv:1907.12455 [cs.DM], 2019.
%F C(n, r) = A051031(n, r) - A068933(n, r).
%F Column k is the inverse Euler transform of column k of A051031. - _Andrew Howroyd_, Mar 10 2020
%e 01: 1;
%e 02: 0, 1;
%e 03: 0, 0, 1;
%e 04: 0, 0, 1, 1;
%e 05: 0, 0, 1, 0, 1;
%e 06: 0, 0, 1, 2, 1, 1;
%e 07: 0, 0, 1, 0, 2, 0, 1;
%e 08: 0, 0, 1, 5, 6, 3, 1, 1;
%e 09: 0, 0, 1, 0, 16, 0, 4, 0, 1;
%e 10: 0, 0, 1, 19, 59, 60, 21, 5, 1, 1;
%e 11: 0, 0, 1, 0, 265, 0, 266, 0, 6, 0, 1;
%e 12: 0, 0, 1, 85, 1544, 7848, 7849, 1547, 94, 9, 1, 1;
%e 13: 0, 0, 1, 0, 10778, 0, 367860, 0, 10786, 0, 10, 0, 1;
%e 14: 0, 0, 1, 509, 88168, 3459383, 21609300, 21609301, 3459386, 88193, 540, 13, 1, 1;
%e 15: 0, 0, 1, 0, 805491, 0, 1470293675, 0, 1470293676, 0, 805579, 0, 17, 0, 1;
%e 16: 0, 0, 1, 4060, 8037418, 2585136675, 113314233808, 733351105934, 733351105935, 113314233813, 2585136741, 8037796, 4207, 21, 1, 1;
%Y Connected regular simple graphs: A005177 (any degree -- sum of rows), this sequence (triangular array), specified degree r (columns): A002851 (r=3), A006820 (r=4), A006821 (r=5), A006822 (r=6), A014377 (r=7), A014378 (r=8), A014381 (r=9), A014382 (r=10), A014384 (r=11).
%Y Triangular arrays C(n,k) counting connected simple k-regular graphs on n vertices with girth *at least* g: this sequence (g=3), A186714 (g=4), A186715 (g=5), A186716 (g=6), A186717 (g=7), A186718 (g=8), A186719 (g=9).
%Y Triangular arrays C(n,k) counting connected simple k-regular graphs on n vertices with girth *exactly* g: A186733 (g=3), A186734 (g=4).
%K nonn,tabl,hard
%O 1,19
%A _David Wasserman_, Mar 08 2002
%E Edited by _Jason Kimberley_, Sep 23 2009, Nov 2011, Jan 2012, and Mar 2012
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