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A323817 Number of connected set-systems covering n vertices with no singletons. 8
1, 0, 1, 12, 1990, 67098648, 144115187673201808, 1329227995784915871895000743748659792, 226156424291633194186662080095093570015284114833799899656335137245499581360 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 30 2019
STATUS
approved

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Last modified February 27 04:33 EST 2024. Contains 370362 sequences. (Running on oeis4.)