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URL: https://oeis.org/A217194

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A217194
Number of unlabeled simple graphs with n nodes of 2 colors whose components are path graphs.
2
1, 2, 6, 16, 42, 106, 267, 656, 1602, 3868, 9270, 22048, 52140, 122580, 286798, 667944, 1549259, 3579738, 8242638, 18917600, 43286909, 98768820, 224768425, 510235760, 1155553468, 2611251662, 5888421059, 13252176464, 29768501556, 66749440076, 149415504274
OFFSET
0,2
COMMENTS
Here, a path graph is a connected graph with no cycles such that each node has degree at most two.
LINKS
FORMULA
G.f.: Product_{i>=1} 1/(1-x^i)^(2^(i-1)+2^(floor((i+1)/2)-1)).
EULER transform of A005418.
EXAMPLE
a(3) = 16 because we have:
w w w; w w b; w b b; b b b;
w w-w; w w-b; w b-b; b w-w; b w-b; b b-b;
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.
MAPLE
with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*(2^(d-1)+
2^(floor((d+1)/2)-1)), d=divisors(j))*a(n-j), j=1..n)/n)
end:
seq(a(n), n=0..30); # Alois P. Heinz, Sep 27 2012
MATHEMATICA
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]
CROSSREFS
Sequence in context: A143123 A102699 A266124 * A304662 A296625 A156664
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Sep 27 2012
STATUS
approved