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A005197
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a(n) = Sum_t t*F(n,t), where F(n,t) (see A033185) is the number of rooted forests with n (unlabeled) nodes and exactly t rooted trees.
(Formerly M2663)
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5
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1, 3, 7, 17, 39, 96, 232, 583, 1474, 3797, 9864, 25947, 68738, 183612, 493471, 1334143, 3624800, 9893860, 27113492, 74577187, 205806860, 569678759, 1581243203, 4400193551, 12273287277, 34307646762, 96093291818, 269654004899, 758014312091, 2134300171031
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OFFSET
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1,2
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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To get a(n), take row n of the triangle in A033185, multiply successive terms by 1, 2, 3, ... and sum. E.g. a(4) = 1*4+2*3+3*1+4*1 = 17.
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MAPLE
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with(numtheory):
t:= proc(n) option remember; local d, j; `if`(n<=1, n,
(add(add(d*t(d), d=divisors(j))*t(n-j), j=1..n-1))/(n-1))
end:
b:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
`if`(min(i, p)<1, 0, add(b(n-i*j, i-1, p-j) *
binomial(t(i)+j-1, j), j=0..min(n/i, p)))))
end:
a:= a-> add(k*b(n, n, k), k=1..n):
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MATHEMATICA
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t[1] = 1; t[n_] := t[n] = Module[{d, j}, Sum[Sum[d*t[d], {d, Divisors[j]}]*t[n-j], {j, 1, n-1}]/(n-1)]; b[1, 1, 1] = 1; b[n_, i_, p_] := b[n, i, p] = If[p>n, 0, If[n == 0, 1, If[Min[i, p]<1, 0, Sum[b[n-i*j, i-1, p-j]*Binomial[t[i]+j-1, j], {j, 0, Min[n/i, p]}]]]]; a[n_] := Sum[k*b[n, n, k], {k, 1, n}]; Table[a[n] // FullSimplify, {n, 1, 30}] (* Jean-François Alcover, Mar 13 2014, after Alois P. Heinz *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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