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A323818
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Number of connected set-systems covering n vertices.
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110
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1, 1, 4, 96, 31840, 2147156736, 9223372011084915712, 170141183460469231602560095199828453376, 57896044618658097711785492504343953923912733397452774312021795134847892828160
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graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,3
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COMMENTS
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Unlike the nearly identical sequence A092918, this sequence does not count under a(1) the a single-vertex hypergraph with no edges.
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LINKS
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FORMULA
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E.g.f.: 1 - x + log(Sum_{n >= 0} 2^(2^n-1) * x^n/n!).
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EXAMPLE
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The a(2) = 4 set-systems:
{{1, 2}}
{{1}, {1,2}}
{{2}, {1,2}}
{{1}, {2}, {1,2}}
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MAPLE
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b:= n-> add(binomial(n, k)*2^(2^(n-k)-1)*(-1)^k, k=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:
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MATHEMATICA
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nn=8;
ser=Sum[2^(2^n-1)*x^n/n!, {n, 0, nn}];
Table[SeriesCoefficient[1-x+Log[ser], {x, 0, n}]*n!, {n, 0, nn}]
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PROG
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(Magma)
m:=12;
f:= func< x | 1-x + Log( (&+[2^(2^n-1)*x^n/Factorial(n): n in [0..m+2]]) ) >;
R<x>:=PowerSeriesRing(Rationals(), m);
(SageMath)
m=12;
def f(x): return 1-x + log(sum(2^(2^n-1)*x^n/factorial(n) for n in range(m+2)))
def A_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( f(x) ).egf_to_ogf().list()
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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