cp's OEIS Frontend

A174145 Number of rooted forests with n nodes in which each component contains at least two nodes.

%I A174145 #34 Aug 12 2016 19:30:10
%S A174145 1,0,1,2,5,11,28,67,171,433,1123,2924,7720,20487,54838,147570,399466,
%T A174145 1086312,2967517,8137552,22395604,61833349,171227674,475442129,
%U A174145 1323449661,3692461865,10324097819,28923331940,81179488039,228240293289,642744665401,1812762839702
%N A174145 Number of rooted forests with n nodes in which each component contains at least two nodes.
%C A174145 Row sums of A174135.
%H A174145 Alois P. Heinz, <a href="/A174145/b174145.txt">Table of n, a(n) for n = 0..1000</a>
%F A174145 a(n) ~ c * d^n / n^(3/2), where d = A051491 = 2.9557652856519949747148..., c = 0.8603881121111431... . - _Vaclav Kotesovec_, Sep 10 2014
%F A174145 In the asymptotics above the constant c = A187770 * (A051491 - 1). - _Vladimir Reshetnikov_, Aug 12 2016
%F A174145 a(n) = A000081(n+1) - A000081(n). - _Vladimir Reshetnikov_, Nov 06 2015
%p A174145 with(numtheory):
%p A174145 t:= proc(n) option remember; local d, j; `if`(n<=1, n,
%p A174145       (add(add(d*t(d), d=divisors(j))*t(n-j), j=1..n-1))/(n-1))
%p A174145     end:
%p A174145 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<2, 0,
%p A174145       add(b(n-i*j, i-1)*binomial(t(i)+j-1, j), j=0..n/i)))
%p A174145     end:
%p A174145 a:= n-> b(n, n):
%p A174145 seq(a(n), n=0..32);  # _Alois P. Heinz_, May 17 2013
%t A174145 t[n_] := t[n] = If[n <= 1, n, Sum[Sum[d*t[d], {d, Divisors[j]}]*t[n-j], {j, 1, n-1}]/(n-1)]; b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<2, 0, Sum[b[n-i*j, i-1]*Binomial[t[i]+j-1, j], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n] // FullSimplify, {n, 0, 32}] (* _Jean-François Alcover_, Mar 19 2014, after _Alois P. Heinz_ *)
%t A174145 t[1] = 1; t[n_] := t[n] = Sum[k t[k] t[n - k m]/(n-1), {k, n-1}, {m, (n-1)/k}]; a[n_] := t[n+1] - t[n]; Table[a[n], {n, 0, 32}] (* _Vladimir Reshetnikov_, Aug 12 2016 *)
%Y A174145 Cf. A000081, A051491, A187770, A255170.
%K A174145 nonn
%O A174145 0,4
%A A174145 _N. J. A. Sloane_, Nov 26 2010