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

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

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

Example

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.

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 *)

Prog

(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

Crossrefs

Row sum of A215977.

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

Sequence in context: A164176 A325860 A217932 * A018086 A004651 A283835

Adjacent sequences: A215975 A215976 A215977 * A215979 A215980 A215981

Keyword

nonn

Author

Alois P. Heinz, Aug 29 2012

Status

approved