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A339642
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Number of rooted trees with n nodes colored using exactly 2 colors.
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3
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0, 2, 10, 44, 196, 876, 4020, 18766, 89322, 431758, 2116220, 10494080, 52569504, 265647586, 1352621168, 6933127446, 35745747902, 185256755454, 964575991660, 5043194697556, 26467075595080, 139375175511598, 736228488297566, 3900073083063348, 20714052518640904
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listen;
history;
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OFFSET
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1,2
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LINKS
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Andrew Howroyd, Table of n, a(n) for n = 1..500
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FORMULA
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a(n) = A038055(n) - 2*A000081(n).
a(n) = 2*(A000151(n) - A000081(n)).
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EXAMPLE
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a(3) = 10 includes 5 trees and their color complements:
(1(12)), (1(22)), (1(1(2))), (1(2(1))), (1(2(2))).
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MAPLE
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b:= proc(n, k) option remember; `if`(n<2, k*n, (add(add(b(d, k)*
d, d=numtheory[divisors](j))*b(n-j, k), j=1..n-1))/(n-1))
end:
a:= n-> b(n, 2)-2*b(n, 1):
seq(a(n), n=1..25); # Alois P. Heinz, Dec 11 2020
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MATHEMATICA
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b[n_, k_] := b[n, k] = If[n < 2, k*n, (Sum[Sum[b[d, k]*d, {d, Divisors[j]}]*b[n - j, k], {j, 1, n - 1}])/(n - 1)];
a[n_] := b[n, 2] - 2*b[n, 1];
Array[a, 25] (* Jean-François Alcover, Jan 04 2021, after Alois P. Heinz *)
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PROG
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(PARI) \\ See A141610 for U(N, m)
seq(n)={U(n, 2) - 2*U(n, 1)}
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CROSSREFS
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Column 2 of A141610.
Cf. A000081, A000151, A038055, A339643.
Sequence in context: A099919 A100397 A084059 * A084609 A105485 A151313
Adjacent sequences: A339639 A339640 A339641 * A339643 A339644 A339645
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KEYWORD
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nonn,changed
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AUTHOR
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Andrew Howroyd, Dec 11 2020
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STATUS
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approved
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