A215979 Number of simple unlabeled graphs on 2*n nodes with exactly n connected components that are trees or cycles.

1, 1, 3, 6, 13, 26, 56, 115, 247, 533, 1175, 2636, 6040, 14078, 33401, 80524, 196897, 487781, 1222279, 3094507, 7905992, 20364597, 52838720, 138001953, 362565398, 957687474, 2542056376, 6777855755, 18146153182, 48766704695, 131517773945, 355842838357

Offset: 0

Author

Alois P. Heinz, Aug 29 2012

Keywords

• nonn

Comments

Limiting sequence of reversed rows of A215977. Also central elements of rows of A215977.

Examples

			a(3) = 6: .o-o o.  .o-o o.  .o-o o.  .o-o o.  .o-o o.  .o o o.
          .| |  .  .|    .  .|\   .  .|/   .  .|    .  .| | |.
          .o-o o.  .o-o o.  .o o o.  .o o-o.  .o o-o.  .o o o.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A215977, A051491.

Programs

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:
g:= proc(n) option remember; local k; `if`(n>2, 1, 0)+ b(n)-
      (add(b(k)*b(n-k), k=0..n) -`if`(irem(n, 2)=0, b(n/2), 0))/2
    end:
p:= proc(n, i, t) option remember; `if`(n p(2*n, 2*n, n):
seq(a(n), n=0..35);

Mathematica

b[n_] := b[n] = If[n <= 1, n, (Sum[DivisorSum[j, #*b[#]&]*b[n-j], {j, 1, n-1}])/(n-1)];
g[n_] := g[n] = If[n > 2, 1, 0] + b[n] - (Sum[b[k]*b[n-k], {k, 0, n}] - If[EvenQ[n], b[n/2], 0])/2;
p[n_, i_, t_] := p[n, i, t] = If[n < t, 0, If[n == t, 1, If[Min[i, t]<1, 0, Sum[Binomial[g[i]+j-1, j]*p[n-i*j, i-1, t-j], {j, 0, Min[n/i, t]}]]]];
a[n_] := p[2*n, 2*n, n];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Feb 26 2017, after Alois P. Heinz *)

Formula

a(n) = A215977(2*n,n).

a(n) ~ c * d^n / n^(5/2), where d = A051491 = 2.9557652856..., c = 4.034813602... . - Vaclav Kotesovec, Aug 31 2014