A005197 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)

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

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

Cf. A000081, A005196, A033185.

Sequence in context: A342600 A229514 A077927 * A147142 A298371 A367396

Adjacent sequences: A005194 A005195 A005196 * A005198 A005199 A005200

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