A215987 -id:A215987 - 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

A215857 Number of simple labeled graphs on n nodes with exactly 7 connected components that are trees or cycles.

Original entry on oeis.org

1, 28, 714, 17220, 424809, 11002068, 303874714, 9016296289, 288135739892, 9913826194272, 366486926833846, 14513217676764534, 613646633464214863, 27609928896732666760, 1317652578222779606269, 66497975770225498765728, 3538905411811229060814213

Offset: 7

Alois P. Heinz, Aug 26 2012

nonn

			a(8) = 28: each graph has one 2-node tree and 6 1-node trees and C(8,2) = 28.

Alois P. Heinz, Table of n, a(n) for n = 7..145

Column k=7 of A215861.

The unlabeled version is A215987.

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, 7):
seq(a(n), n=7..25);

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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