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A292554
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Number of rooted unlabeled trees on n nodes where each node has at most 9 children.
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11
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1, 1, 1, 2, 4, 9, 20, 48, 115, 286, 719, 1841, 4763, 12477, 32947, 87735, 235162, 634212, 1719325, 4683368, 12810871, 35177357, 96926335, 267909285, 742641309, 2064029034, 5750500663, 16057186086, 44929879114, 125962026154, 353773417487, 995269027339
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OFFSET
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0,4
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LINKS
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FORMULA
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Functional equation of G.f. is T(z) = z + z*Sum_{q=1..9} Z(S_q)(T(z)) with Z(S_q) the cycle index of the symmetric group. Alternate FEQ is
T(z) = 1 + z*Z(S_9)(T(z)).
a(n) / a(n+1) ~ 0.338343552789108712866488147828528012266693326385052387884853... - Robert A. Russell, Feb 11 2023
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MAPLE
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b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
`if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
b(n-i*j, i-1, t-j, k), j=0..min(t, n/i))))
end:
a:= n-> `if`(n=0, 1, b(n-1$2, 9$2)):
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MATHEMATICA
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b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, 1, If[i < 1, 0, Sum[ Binomial[b[i - 1, i - 1, k, k] + j - 1, j]*b[n - i*j, i - 1, t - j, k], {j, 0, Min[t, n/i]}]]];
a[n_] := If[n == 0, 1, b[n - 1, n - 1, 9, 9]];
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CROSSREFS
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Cf. A000081, A001190, A000598, A036718, A036721, A036722, A182378, A244372, A292553, A292555, A292556.
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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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