A215978 Number of simple unlabeled graphs on n nodes with connected components that are trees or cycles.

1, 1, 2, 4, 8, 14, 28, 52, 104, 206, 429, 903, 1982, 4430, 10206, 23966, 57522, 140236, 347302, 870682, 2207819, 5651437, 14590703, 37948338, 99358533, 261684141, 692906575, 1843601797, 4926919859, 13220064562, 35604359531, 96218568474, 260850911485

Offset: 0

Author

Alois P. Heinz, Aug 29 2012

Keywords

• nonn

Examples

			a(4) = 8:
.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..1000

Crossrefs

Row sum of A215977.

The labeled version is A144164. The inverse Euler transform is A215981.

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) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(g(i)+j-1,j)*p(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> p(n, n):
seq(a(n), n=0..40);

Mathematica

b[n_] := b[n] = If[n <= 1, n, (Sum[Sum[d*b[d], {d, Divisors[j]}]*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[Mod[n, 2] == 0, b[n/2], 0])/2;
p[n_, i_] := p[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[g[i] + j - 1, j]*p[n - i*j, i - 1], {j, 0, n/i}]]];
a[n_] := p[n, n];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 21 2017, translated from Maple *)

Python

from sympy.core.cache import cacheit
from sympy import binomial, divisors
@cacheit
def b(n): return n if n<2 else sum([sum([d*b(d) for d in divisors(j)])*b(n - j) for j in range(1, n)])//(n - 1)
@cacheit
def g(n): return (1 if n>2 else 0) + b(n) - (sum([b(k)*b(n - k) for k in range(n + 1)]) - (b(n//2) if n%2==0 else 0))//2
@cacheit
def p(n, i): return 1 if n==0 else 0 if i<1 else sum([binomial(g(i) + j - 1, j)*p(n - i*j, i - 1) for j in range(n//i + 1)])
def a(n): return p(n, n)
print([a(n) for n in range(41)]) # Indranil Ghosh, Aug 07 2017

Formula

a(n) = Sum_{k=0..n} A215977(n,k).

a(n) ~ c * d^n / n^(5/2), where d = A051491 = 2.95576528565199497471481752412... is Otter's rooted tree constant, and c = 1.085767435235426664262830616636... . - Vaclav Kotesovec, Mar 22 2017