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

0,3

Comments

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

Links

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

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

Example

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.

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<t, 0, `if`(n=t, 1,
      `if`(min(i, t)<1, 0, add(binomial(g(i)+j-1, j)*
       p(n-i*j, i-1, t-j), j=0..min(n/i, t)))))
    end:
a:= 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 *)

Crossrefs

Cf. A215977, A051491.

Sequence in context: A215988 A215989 A215980 * A273226 A291726 A280563

Adjacent sequences: A215976 A215977 A215978 * A215980 A215981 A215982

Keyword

nonn

Author

Alois P. Heinz, Aug 29 2012

Status

approved