This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A331563 #37 Feb 16 2025 08:33:59 %S A331563 0,0,20,1610,129654,11688369,1194822915,137766789810,17758192128830, %T A331563 2535895233070628,397875362655895761,68087081506276861665, %U A331563 12626853606957534296975,2523446241515288646389325 %N A331563 Number of labeled cyclic graphs with n edges and 2n vertices. %H A331563 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PawGraph.html">Paw Graph</a> %F A331563 a(n) = A331505(2n) - A302112(n). %e A331563 a(4) = 1610 since we have 3 non-isomorphic cyclic graphs with 4 edges and 8 nodes. (See illustration below.) %e A331563 To compute a(4) we can consult A057500, which provides the number of labeled connected unicycles. Because A057500(4)=15, and knowing that there are 3 labeled squares, we have 15-3 = 12 Paw Graphs [see Weisstein link]. So graph 1 is labeled in 12 * C(8,4) = 840 ways. Graph 2 is labeled in 3* C(8,4) = 210 ways. A105599 gives 10 as the number of labeled forests with 5 nodes and 4 components, so graph 3 is labeled in 10 * C(8,3) = 560 ways. We have 840 + 210 + 560 = 1610. %e A331563 . %e A331563 graph 1 graph 2 graph 3 (triangle + forest with %e A331563 5 nodes and 4 components) %e A331563 *--* *--* *--* * %e A331563 | /| | | | / | %e A331563 |/ | | | |/ | %e A331563 * * *--* * * %e A331563 * * * * * * * * * * * %Y A331563 Cf. A331505, A302112, A057500, A105599. %K A331563 nonn %O A331563 1,3 %A A331563 _Washington Bomfim_, Jan 20 2020