A323817 Number of connected set-systems covering n vertices with no singletons.

1, 0, 1, 12, 1990, 67098648, 144115187673201808, 1329227995784915871895000743748659792, 226156424291633194186662080095093570015284114833799899656335137245499581360

Offset

0,4

Links

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

Formula

Logarithmic transform of A323816.

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}}
The A323816(4) - a(4) = 3 disconnected set-systems covering n vertices with no singletons:
  {{1, 2}, {3, 4}}
  {{1, 3}, {2, 4}}
  {{1, 4}, {2, 3}}

Maple

b:= n-> add(2^(2^(n-j)-n+j-1)*binomial(n, j)*(-1)^j, j=0..n):
a:= proc(n) option remember; b(n)-`if`(n=0, 0, add(
       k*binomial(n, k)*b(n-k)*a(k), k=1..n-1)/n)
    end:
seq(a(n), n=0..8);  # Alois P. Heinz, Jan 30 2019

Mathematica

nn=10;
ser=Sum[Sum[(-1)^(n-k)*Binomial[n, k]*2^(2^k-k-1), {k, 0, n}]*x^n/n!, {n, 0, nn}];
Table[SeriesCoefficient[1+Log[ser], {x, 0, n}]*n!, {n, 0, nn}]

Prog

(Magma)
m:=10;
A323816:= func< n | (&+[(-1)^(n-j)*Binomial(n, j)*2^(2^j -j-1): j in [0..n]]) >;
f:= func< x | 1 + Log((&+[A323816(j)*x^j/Factorial(j): j in [0..m+2]])) >;
R<x>:=PowerSeriesRing(Rationals(), m+1);
Coefficients(R!(Laplace( f(x) ))); // G. C. Greubel, Oct 05 2022
(SageMath)
m=10
def A323816(n): return sum((-1)^j*binomial(n, j)*2^(2^(n-j) -n+j-1) for j in range(n+1))
def A323817_list(prec):
    P.<x> = PowerSeriesRing(QQ, prec)
    return P( 1 + log(sum(A323816(j)*x^j/factorial(j) for j in range(m+2))) ).egf_to_ogf().list()
A323817_list(m) # G. C. Greubel, Oct 05 2022

Crossrefs

Cf. A001187, A016031, A048143, A092918, A293510, A317795, A323816 (not necessarily connected), A323818 (with singletons), A323819, A323820 (unlabeled case).

Sequence in context: A326601 A265216 A011920 * A263584 A323816 A208252

Adjacent sequences: A323814 A323815 A323816 * A323818 A323819 A323820

Keyword

nonn

Author

Gus Wiseman, Jan 30 2019

Status

approved