%I #22 Mar 03 2024 09:29:28

%S 1,2,2,6,12,33,93,287,940,3309,12183,47133,190061,796405,3456405,

%T 15501183,71681170,341209173,1669411182,8384579797,43180474608,

%U 227797465130,1229915324579,6790642656907,38311482445514,220712337683628,1297542216770482,7779452884747298

%N Number of connected graphs with n edges with loops allowed.

%C Inverse Euler transform of A053419.

%C From _R. J. Mathar_, Jul 25 2017: (Start)

%C The Multiset Transform gives the number of graphs with n edges (loops allowed) and k components (0<=k<=n):

%C 1

%C 0 2

%C 0 2 3

%C 0 6 4 4

%C 0 12 15 6 5

%C 0 33 36 24 8 6

%C 0 93 111 64 33 10 7

%C 0 287 324 207 92 42 12 8

%C 0 940 1036 633 308 120 51 14 9

%C 0 3309 3408 2084 966 409 148 60 16 10

%C 0 12183 11897 6959 3243 1305 510 176 69 18 11

%C 0 47133 43137 24415 10970 4432 1644 611 204 78 20 12

%C 0 190061 163608 88402 38763 15125 5628 1983 712 232 87 22 13

%C 0 796405 644905 332979 140671 53732 19316 6824 2322 813 260 96 24 14

%C 0 3456405 2639871 1299054 529179 195517 68878 23515 8020 2661 914 288 105 26 15 (End)

%H Andrew Howroyd, <a href="/A191970/b191970.txt">Table of n, a(n) for n = 0..50</a>

%H Gus Wiseman, <a href="/A191970/a191970.png">Non-isomorphic representatives of the a(1) = 1 through a(4) = 12 connected graphs with loops.</a>

%e a(1)=2: Either one node with the edge equal to a loop, or two nodes connected by the edge. a(2)=2: Either three nodes on a chain connected by the two edges, or two nodes connected by an edge, one node with a loop. Apparently multi-loops are not allowed (?). - _R. J. Mathar_, Jul 25 2017

%o (PARI) \\ See A322114 for InvEulerMT, G.

%o seq(n)={vecsum([Vec(p+O(y^n), -n) | p<-InvEulerMT(vector(n, k, G(k, y + O(y^n))))])} \\ _Andrew Howroyd_, Oct 22 2019

%Y Row sums of A322114.

%Y Cf. A000664, A002905, A007718, A050535, A053419, A054923, A191646, A191970, A275421, A322133, A322151, A322152.

%K nonn

%O 0,2

%A _Alberto Tacchella_, Jun 20 2011

%E Terms a(25) and beyond from _Andrew Howroyd_, Oct 22 2019