A215930 Number of forests on unlabeled nodes with n edges and no single node trees.

1, 1, 2, 4, 8, 16, 34, 71, 154, 341, 768, 1765, 4134, 9838, 23766, 58226, 144353, 361899, 916152, 2339912, 6023447, 15617254, 40752401, 106967331, 282267774, 748500921, 1993727506, 5332497586, 14316894271, 38574473086, 104273776038, 282733466684, 768809041078

Offset

0,3

Comments

Each forest counted by a(n) with n>0 has number of nodes from the interval [n+1,2*n] and number of trees in [1,n].

Also limiting sequence of reversed rows of A095133.

Differs from A011782 first at n=6 (32) and from A088325 at n=8 (153).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..650

Formula

a(n) = A095133(2*n,n).

a(n) = A105821(2*n+1,n+1). - Alois P. Heinz, Jul 10 2013

a(n) = A136605(2*n+1,n). - Alois P. Heinz, Apr 11 2014

a(n) ~ c * d^n / n^(5/2), where d = A051491 = 2.955765285..., c = 3.36695186... . - Vaclav Kotesovec, Sep 10 2014

Example

a(0) = 1: (  ), the empty forest with 0 trees and 0 edges.
a(1) = 1: ( o-o ), 1 tree and 1 edge.                      o
a(2) = 2: ( o-o-o ), ( o-o o-o ).                          |
a(3) = 4: ( o-o-o-o ), ( o-o-o o-o ), ( o-o o-o o-o ), ( o-o-o ).

Maple

with(numtheory):
b:= proc(n) option remember; local d, j; `if`(n<=1, n,
      (add(add(d*b(d), d=divisors(j)) *b(n-j), j=1..n-1))/(n-1))
    end:
t:= proc(n) option remember; local k; `if` (n=0, 1, b(n)-
      (add(b(k)*b(n-k), k=0..n)-`if`(irem(n, 2)=0, b(n/2), 0))/2)
    end:
g:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
      `if`(min(i, p)<1, 0, add(g(n-i*j, i-1, p-j)*
       binomial(t(i)+j-1, j), j=0..min(n/i, p)))))
    end:
a:= n-> g(2*n, 2*n, n):
seq(a(n), n=0..40);

Mathematica

nn = 30; t[x_] := Sum[a[n] x^n, {n, 1, nn}]; a[0] = 0;
a[1] = 1; sol =
SolveAlways[
  0 == Series[
    t[x] - x Product[1/(1 - x^i)^a[i], {i, 1, nn}], {x, 0, nn}], x];
b[x_] := Sum[a[n] x^n /. sol, {n, 0, nn}]; ft =
Drop[Flatten[
   CoefficientList[Series[b[x] - (b[x]^2 - b[x^2])/2, {x, 0, nn}],
    x]], 1]; Drop[
CoefficientList[
  Series[Product[1/(1 - y ^(i - 1))^ft[[i]], {i, 2, nn}], {y, 0, nn}],
y], -1] (* Geoffrey Critzer, Nov 10 2014 *)

Crossrefs

Cf. A000055, A005195, A011782, A051491, A088325, A095133, A105821, A136605.

Sequence in context: A275443 A288170 A088325 * A367660 A288260 A006210

Adjacent sequences: A215927 A215928 A215929 * A215931 A215932 A215933

Keyword

nonn

Author

Alois P. Heinz, Aug 27 2012

Status

approved