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A319369
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Number of series-reduced rooted trees with n leaves of n colors.
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4
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1, 3, 28, 430, 9376, 269675, 9632960, 411395268, 20445999734, 1159248404721, 73846864163348, 5221802726902476, 405858598184643930, 34392275731729465799, 3155760058245300968416, 311720334688779807141832, 32980137195294216968253900, 3720954854814866649904474180
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OFFSET
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1,2
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COMMENTS
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Not all of the n colors need to be used.
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LINKS
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FORMULA
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a(n) ~ c * d^n * n^(n - 3/2), where d = 1/(2*log(2) - 1) = 2.588699449562089830805384431942090... and c = 0.2580000331300831455241033648... - Vaclav Kotesovec, Sep 18 2019, updated Mar 16 2024
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MAPLE
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b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(binomial(A(i, k)+j-1, j)*b(n-i*j, i-1, k), j=0..n/i)))
end:
A:= (n, k)-> `if`(n<2, n*k, b(n, n-1, k)):
a:= n-> A(n$2):
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MATHEMATICA
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b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[A[i, k] + j - 1, j]*b[n - i*j, i - 1, k], {j, 0, n/i}]]];
A[n_, k_] := If[n < 2, n*k, b[n, n - 1, k]];
a[n_] := A[n, n];
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PROG
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(PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
a(n)={my(v=[n]); for(n=2, n, v=concat(v, EulerT(concat(v, [0]))[n])); v[n]}
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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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