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A035055 Number of forests of different trees. 1
1, 1, 1, 2, 3, 6, 12, 24, 49, 105, 231, 517, 1188, 2783, 6643, 16101, 39606, 98605, 248287, 631214, 1618878, 4183964, 10889305, 28517954, 75111521, 198851386, 528929895, 1412993746, 3789733399, 10201625514, 27555373561, 74664487653, 202908119046, 552939614498 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

N. J. A. Sloane, Transforms

FORMULA

Weigh transform of A000055.

a(n) ~ c * d^n / n^(5/2), where d = A051491 = 2.9557652856519949747148175..., c = 0.89246007934060351292465521837... . - Vaclav Kotesovec, Aug 25 2014

MAPLE

with(numtheory):

b:= proc(n) option remember; `if`(n<2, n,

      (add(add(d*b(d), d=divisors(j))*b(n-j), j=1..n-1))/(n-1))

    end:

h:= proc(n) option remember; `if`(n=0, 1, b(n)-(add(b(k)*b(n-k),

      k=0..n) -`if`(irem(n, 2)=0, b(n/2), 0))/2)

    end:

g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

      add(binomial(h(i), j)*g(n-i*j, i-1), j=0..n/i)))

    end:

a:= n-> g(n, n):

seq(a(n), n=0..40); # Alois P. Heinz, May 19 2013

MATHEMATICA

nn = 20; t[x_] := Sum[a[n] x^n, {n, 1, nn}]; a[0] = 0;

b = Flatten[

  sol = SolveAlways[

    0 == Series[

      t[x] - x Product[1/(1 - x^i)^ a[i], {i, 1, nn}], {x, 0, nn}],

    x]; Table[a[n], {n, 0, nn}] /. sol];

r[x_] := Sum[b[[n]] x^(n - 1), {n, 1, nn + 1}]; c =

Drop[CoefficientList[

   Series[r[x] - (r[x]^2/2 - r[x^2]/2), {x, 0, nn}], x],

  1]; CoefficientList[

Series[Product[(1 + x^i)^c[[i]], {i, 1, nn}], {x, 0, nn}], x] (* Geoffrey Critzer, Nov 15 2014 *)

CROSSREFS

Cf. A005195.

Sequence in context: A098011 A110164 A042950 * A119559 A045761 A187741

Adjacent sequences:  A035052 A035053 A035054 * A035056 A035057 A035058

KEYWORD

nonn

AUTHOR

Christian G. Bower, Oct 15 1998

STATUS

approved

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Last modified November 20 14:54 EST 2019. Contains 329337 sequences. (Running on oeis4.)