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A215978 Number of simple unlabeled graphs on n nodes with connected components that are trees or cycles. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
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
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Aug 29 2012
STATUS
approved

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Last modified July 12 17:26 EDT 2024. Contains 374251 sequences. (Running on oeis4.)