A217194 - OEIS

%I #20 May 17 2018 16:26:43

%S 1,2,6,16,42,106,267,656,1602,3868,9270,22048,52140,122580,286798,

%T 667944,1549259,3579738,8242638,18917600,43286909,98768820,224768425,

%U 510235760,1155553468,2611251662,5888421059,13252176464,29768501556,66749440076,149415504274

%N Number of unlabeled simple graphs with n nodes of 2 colors whose components are path graphs.

%C Here, a path graph is a connected graph with no cycles such that each node has degree at most two.

%H Alois P. Heinz, <a href="/A217194/b217194.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: Product_{i>=1} 1/(1-x^i)^(2^(i-1)+2^(floor((i+1)/2)-1)).

%F EULER transform of A005418.

%e a(3) = 16 because we have:

%e w w w; w w b; w b b; b b b;

%e w w-w; w w-b; w b-b; b w-w; b w-b; b b-b;

%e w-w-w; w-w-b; w-b-w; b-w-b; b-b-w; b-b-b, where the 2 colors are black b and white w.

%p with(numtheory):

%p a:= proc(n) option remember; `if`(n=0, 1, add(add(d*(2^(d-1)+

%p       2^(floor((d+1)/2)-1)), d=divisors(j))*a(n-j), j=1..n)/n)

%p     end:

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, Sep 27 2012

%t nn=30;p=Product[1/(1- x^i)^(2^(i-1)+2^(Floor[(i+1)/2]-1)),{i,1,nn}];CoefficientList[Series[p,{x,0,nn}],x]

%Y Cf.: A217093, A005380.

%K nonn

%O 0,2

%A _Geoffrey Critzer_, Sep 27 2012