A215982 -id:A215982 - cp's OEIS frontend

A215977 Number T(n,k) of simple unlabeled graphs on n nodes with exactly k connected components that are trees or cycles; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 3, 3, 1, 1, 0, 4, 5, 3, 1, 1, 0, 7, 10, 6, 3, 1, 1, 0, 12, 17, 12, 6, 3, 1, 1, 0, 24, 33, 23, 13, 6, 3, 1, 1, 0, 48, 62, 47, 25, 13, 6, 3, 1, 1, 0, 107, 127, 92, 53, 26, 13, 6, 3, 1, 1, 0, 236, 267, 189, 106, 55, 26, 13, 6, 3, 1, 1

Offset: 0

Alois P. Heinz, Aug 29 2012

			T(4,1) = 3: .o-o.  .o-o.  .o-o.
            .| |.  .|  .  .|\ .
            .o-o.  .o-o.  .o o.
.
T(4,2) = 3: .o-o.  .o-o.  .o-o.
            .|/ .  .|  .  .   .
            .o o.  .o o.  .o-o.
.
T(5,1) = 4: .o-o-o.  .o-o-o.  .o-o-o.  .o-o-o.
            .|  / .  .|    .  .| |  .  . /|  .
            .o-o  .  .o-o  .  .o o  .  .o o  .
.
T(5,2) = 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  .
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  1,  1;
  0,  2,  1,  1;
  0,  3,  3,  1,  1;
  0,  4,  5,  3,  1,  1;
  0,  7, 10,  6,  3,  1,  1;
  0, 12, 17, 12,  6,  3,  1,  1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-10 give: A000007, A215981, A215982, A215983, A215984, A215985, A215986, A215987, A215988, A215989, A215980.
Row sums give: A215978.
Limiting sequence of reversed rows gives: A215979.
The labeled version of this triangle is A215861.

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[nJean-François Alcover, Dec 04 2014, after Alois P. Heinz *)

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 nIndranil Ghosh, Aug 07 2017

A215852 Number of simple labeled graphs on n nodes with exactly 2 connected components that are trees or cycles.

Original entry on oeis.org

1, 3, 19, 135, 1267, 15029, 218627, 3783582, 75956664, 1734309929, 44357222772, 1255715827483, 38971877812380, 1315634598619830, 47994245894462576, 1881406032047006812, 78870928008704884848, 3520953336130828001295, 166762291211479030734580

Offset: 2

Alois P. Heinz, Aug 25 2012

nonn

			a(3) = 3:
.1 2.  .1-2.  .1 2.
.|. .  . . .  . / .
.3...  .3...  .3...

Alois P. Heinz, Table of n, a(n) for n = 2..145

Column k=2 of A215861.

The unlabeled version is A215982.

T:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
      `if`(n=0, 1, add(binomial(n-1, i)*T(n-1-i, k-1)*
      `if`(i<2, 1, i!/2 +(i+1)^(i-1)), i=0..n-k)))
    end:
a:= n-> T(n, 2):
seq(a(n), n=2..25);

T[n_, k_]:=T[n, k]=If[k<0 || k>n, 0, If[n==0, 1, Sum[Binomial[n - 1, i] T[n - 1 - i, k - 1] If[i<2, 1, i!/2 + (i + 1)^(i - 1)], {i, 0, n - k}]]]; Table[T[n, 2], {n, 2, 50}] (* Indranil Ghosh, Aug 07 2017, after Maple *)

from sympy.core.cache import cacheit
from sympy import binomial, factorial as f
@cacheit
def T(n, k): return 0 if k<0 or k>n else 1 if n==0 else sum([binomial(n - 1, i)*T(n - 1 - i, k - 1)*(1 if i<2 else f(i)//2 + (i + 1)**(i - 1)) for i in range(n - k + 1)])
def a(n): return T(n , 2)
print([a(n) for n in range(2, 51)]) # Indranil Ghosh, Aug 07 2017, after maple code

a(n) ~ c * n^(n-2), where c = 0.511564031298... . - Vaclav Kotesovec, Sep 07 2014

cp's OEIS Frontend

A215977 Number T(n,k) of simple unlabeled graphs on n nodes with exactly k connected components that are trees or cycles; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

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