A105599 -id:A105599 - cp's OEIS frontend

A218691 Number of ways to paint some (possibly none or all) of the trees over all forests on n labeled nodes.

1, 2, 6, 26, 156, 1242, 12616, 158034, 2372880, 41725106, 843126624, 19277549898, 492447987136, 13907344659210, 430397513894016, 14487404695687298, 527023721684738304, 20605894357093102434, 861761850029367846400, 38387125875316048363386, 1814541564588778500135936

Offset: 0

Geoffrey Critzer, Nov 04 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..150

Cf. A101313.

T:= -LambertW(-x):
egf:= exp(T-T^2/2)^2:
a:= n-> n! * coeff(series(egf, x, n+1), x, n):
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 04 2012

nn=20;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Range[0,nn]!CoefficientList[ Series[Exp[(t-t^2/2)]^2,{x,0,nn}],x]

E.g.f.: exp(T(x) - T(x)^2/2)^2 where T(x) is e.g.f. for A000169.
a(n) = Sum_{m=1..n} A105599(n,m)*2^m.
a(n) ~ 2*n^(n-2)*exp(1). - Vaclav Kotesovec, Aug 16 2013

A331563 Number of labeled cyclic graphs with n edges and 2n vertices.

Original entry on oeis.org

0, 0, 20, 1610, 129654, 11688369, 1194822915, 137766789810, 17758192128830, 2535895233070628, 397875362655895761, 68087081506276861665, 12626853606957534296975, 2523446241515288646389325

Offset: 1

Washington Bomfim, Jan 20 2020

nonn

			a(4) = 1610 since we have 3 non-isomorphic cyclic graphs with 4 edges and 8 nodes. (See illustration below.)
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.
.
  graph 1    graph 2    graph 3 (triangle + forest with
                                 5 nodes and 4 components)
   *--*       *--*       *--* *
   | /|       |  |       | /  |
   |/ |       |  |       |/   |
   *  *       *--*       *    *
  * * * *    * * * *      * * *

Eric Weisstein's World of Mathematics, Paw Graph

Cf. A331505, A302112, A057500, A105599.

a(n) = A331505(2n) - A302112(n).

cp's OEIS Frontend

A218691 Number of ways to paint some (possibly none or all) of the trees over all forests on n labeled nodes.

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