A005198 a(n) is the number of forests with n (unlabeled) nodes in which each component tree is planted, that is, is a rooted tree in which the root has degree 1.

0, 1, 1, 3, 5, 13, 27, 68, 160, 404, 1010, 2604, 6726, 17661, 46628, 124287, 333162, 898921, 2437254, 6640537, 18166568, 49890419, 137478389, 380031868, 1053517588, 2928246650, 8158727139, 22782938271, 63752461474, 178740014515, 502026565792, 1412409894224

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

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..2136 (first 118 terms from Washington Bomfim)
• E. M. Palmer and A. J. Schwenk, On the number of trees in a random forest, J. Combin. Theory, B 27 (1979), 109-121.

Crossrefs

Cf. A000081.

Programs

Maple

g:= proc(n) option remember; `if`(n<=1, n, (add(add(d*g(d),
       d=numtheory[divisors](j))*g(n-j), j=1..n-1))/(n-1))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<2, 0, add(
       binomial(g(i-1)+j-1, j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=1..40);  # Alois P. Heinz, Jul 07 2020

Mathematica

g[n_] := g[n] = If[n <= 1, n, Sum[Sum[d g[d], {d, Divisors[j]}] g[n - j], {j, 1, n - 1}]/(n - 1)];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 2, 0, Sum[Binomial[g[i - 1] + j - 1, j] b[n - i j, i - 1], {j, 0, n/i}]]];
a[n_] := b[n, n];
Array[a, 40] (* Jean-François Alcover, Nov 08 2020, after Alois P. Heinz *)

PARI

g(m) = {my(f); if(m==0, return(1)); f = vector(m+1); f[1]=1;
for(j=1, m, f[j+1]=1/j * sum(k=1, j, sumdiv(k,d, d * f[d]) * f[j-k+1])); f[m+1] };
global(max_n = 130); A000081 = vector(max_n, n, g(n-1));
seq(n)={my(s=0, D, c, P_1); if(n==1,return(0)); forpart(P_1 = n, D = Set(P_1); c = vector(#D); for(k=1, #D, c[k] = #select(x->x == D[k], Vec(P_1)));
s += prod(k=1, #D, binomial( A000081[D[k]-1] + c[k] - 1, c[k]) ),[2,n],[1,n]); s}; \\ Washington Bomfim, Jul 05 2020

Formula

a(1) = 0, if n >= 2 a(n) = Sum_{P_1(n)}( Product_{k=2..n} binomial(A000081(k-1) + c_k - 1, c_k) ), where P_1(n) are the partitions of n without parts equal to 1: 2*c_2 + ... + n*c_n = n; c_2, ..., c_n >= 0. - Washington Bomfim, Jul 05 2020

Extensions

Definition clarified and more terms added from Palmer-Schwenk by N. J. A. Sloane, May 29 2012