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A215982 Number of simple unlabeled graphs on n nodes with exactly 2 connected components that are trees or cycles. 3
1, 1, 3, 5, 10, 17, 33, 62, 127, 267, 587, 1326, 3085, 7326, 17731, 43585, 108563, 273544, 696113, 1787042, 4623125, 12043071, 31565842, 83200763, 220413272, 586625403, 1567930743, 4207181144, 11329835687, 30613313339, 82975300030, 225552632043, 614787508640 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 2..650

FORMULA

a(n) ~ c * d^n / n^(5/2), where d = A051491 = 2.9557652856519949747148..., c = 0.3339525664158379... . - Vaclav Kotesovec, Sep 07 2014

EXAMPLE

a(5) = 5: .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(n, n, 2):

seq(a(n), n=2..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_, 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[n, n, 2] // FullSimplify; Table[a[n], {n, 2, 40}] // Flatten (* Jean-Fran├žois Alcover, Dec 04 2014, 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, t): return 0 if n<t else 1 if n==t else 0 if min(i, t)<1 else sum([binomial(g(i) + j - 1, j)*p(n - i*j, i - 1, t - j) for j in range(min(n//i, t) + 1)])

def a(n): return p(n, n, 2)

print([a(n) for n in range(2, 41)]) # Indranil Ghosh, Aug 07 2017

CROSSREFS

Column k=2 of A215977.

The labeled version is A215852.

Sequence in context: A054166 A265317 A054157 * A026621 A096395 A026687

Adjacent sequences:  A215979 A215980 A215981 * A215983 A215984 A215985

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Aug 29 2012

STATUS

approved

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Last modified July 1 09:41 EDT 2022. Contains 354958 sequences. (Running on oeis4.)