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A339643
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Number of rooted trees with n nodes colored using exactly 3 colors.
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3
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0, 0, 9, 102, 870, 6744, 50421, 371676, 2731569, 20113005, 148752507, 1106207331, 8274878880, 62263100994, 471138360426, 3584051515209, 27399942354822, 210432444531798, 1622954350900455, 12565580096217270, 97634810663895132, 761110656740387865, 5951117699678438271
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listen;
history;
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OFFSET
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1,3
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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) = A038059(n) - 3*A038055(n) + 3*A000081(n).
a(n) = 3*(A006964(n) - 2*A000151(n) + A000081(n)).
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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, 3)-3*b(n, 2)+3*b(n, 1):
seq(a(n), n=1..23); # 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, 3] - 3 b[n, 2] + 3 b[n, 1];
Array[a, 23] (* 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, 3) - 3*U(n, 2) + 3*U(n, 1)}
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CROSSREFS
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Column 3 of A141610.
Cf. A000081, A000151, A006964, A038055, A038059, A339642.
Sequence in context: A269732 A007133 A183518 * A083452 A081461 A231646
Adjacent sequences: A339640 A339641 A339642 * A339644 A339645 A339646
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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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