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. (Formerly M2491)

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

1,4

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.

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

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 *)

Prog

(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

Crossrefs

Cf. A000081.

Sequence in context: A000631 A026569 A035082 * A160823 A077443 A147196

Adjacent sequences: A005195 A005196 A005197 * A005199 A005200 A005201

Keyword

nonn

Author

N. J. A. Sloane

Extensions

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

Status

approved