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A215981
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Number of simple unlabeled graphs on n nodes with exactly 1 connected component that is a tree or a cycle.
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4
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1, 1, 2, 3, 4, 7, 12, 24, 48, 107, 236, 552, 1302, 3160, 7742, 19321, 48630, 123868, 317956, 823066, 2144506, 5623757, 14828075, 39299898, 104636891, 279793451, 751065461, 2023443033, 5469566586, 14830871803, 40330829031, 109972410222, 300628862481
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OFFSET
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1,3
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LINKS
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FORMULA
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a(1) = a(2) = 1, a(n) = 1 + A000055(n) for n>=3.
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EXAMPLE
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a(5) = 4: .o-o-o. .o-o-o. .o-o-o. .o-o-o.
.| / . .| . .| | . . /| .
.o-o . .o-o . .o o . .o o .
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MAPLE
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with(numtheory):
b:= proc(n) option remember; local d, j; `if`(n<=1, n,
(add(add(d*b(d), d=divisors(j)) *b(n-j), j=1..n-1))/(n-1))
end:
a:= proc(n) option remember; local k; `if`(n>2, 1, 0)+ b(n)-
(add(b(k)*b(n-k), k=0..n) -`if`(irem(n, 2)=0, b(n/2), 0))/2
end:
seq(a(n), n=1..40);
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MATHEMATICA
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b[n_] := b[n] = If[n <= 1, n, (Sum[Sum[d*b[d], {d, Divisors[j]}]*b[n - j], {j, 1, n - 1}])/(n - 1)];
a[n_] := a[n] = If[n > 2, 1, 0] + b[n] - (Sum[b[k]*b[n - k], {k, 0, n}] - If[Mod[n, 2] == 0, b[n/2], 0])/2;
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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