OFFSET
1,2
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..600
E. M. Palmer and A. J. Schwenk, On the number of trees in a random forest, J. Combin. Theory, B 27 (1979), 109-121.
FORMULA
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.
a(n) ~ c * d^n / n^(3/2), where d = A051491 = 2.955765285..., c = 2.85007275... . - Vaclav Kotesovec, Sep 10 2014
MAPLE
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):
seq(a(n), n=1..40); # Alois P. Heinz, Aug 20 2012
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane. Definition clarified by N. J. A. Sloane, May 29 2012
EXTENSIONS
More terms from Alois P. Heinz, Aug 20 2012
STATUS
approved