A323816 Number of set-systems covering n vertices with no singletons.

1, 0, 1, 12, 1993, 67098768, 144115187673233113, 1329227995784915871895000745158568460, 226156424291633194186662080095093570015284114833799899660370362545578585265

Offset

0,4

Links

G. C. Greubel, Table of n, a(n) for n = 0..11

Formula

Inverse binomial transform of A016031 shifted once to the left.

Example

The a(3) = 12 set-systems:
  {{1,2,3}}
  {{1,2}, {1,3}}
  {{1,2}, {2,3}}
  {{1,3}, {2,3}}
  {{1,2}, {1,2,3}}
  {{1,3}, {1,2,3}}
  {{2,3}, {1,2,3}}
  {{1,2}, {1,3}, {2,3}}
  {{1,2}, {1,3}, {1,2,3}}
  {{1,2}, {2,3}, {1,2,3}}
  {{1,3}, {2,3}, {1,2,3}}
  {{1,2}, {1,3}, {2,3}, {1,2,3}}

Maple

a:= n-> add(2^(2^(n-j)-n+j-1)*binomial(n, j)*(-1)^j, j=0..n):
seq(a(n), n=0..8);  # Alois P. Heinz, Jan 30 2019

Mathematica

Table[Sum[(-1)^(n-k)*Binomial[n, k]*2^(2^k-k-1), {k, 0, n}], {n, 0, 8}]

Prog

(Magma)
[(&+[(-1)^(n-j)*Binomial(n, j)*2^(2^j -j-1): j in [0..n]]): n in [0..12]]; // G. C. Greubel, Oct 05 2022
(SageMath)
def A323816(n): return sum((-1)^j*binomial(n, j)*2^(2^(n-j) -n+j-1) for j in range(n+1))
[A323816(n) for n in range(12)] # G. C. Greubel, Oct 05 2022

Crossrefs

Cf. A000295, A000371, A000612, A003465 (with singletons), A006129 (covers by pairs), A016031, A055154, A055621, A305001, A317795 (unlabeled case), A323817 (connected case).

Sequence in context: A011920 A323817 A263584 * A208252 A204622 A369336

Adjacent sequences: A323813 A323814 A323815 * A323817 A323818 A323819

Keyword

nonn

Author

Gus Wiseman, Jan 30 2019

Status

approved