A323817 - OEIS

%I #10 Oct 06 2022 08:17:20

%S 1,0,1,12,1990,67098648,144115187673201808,

%T 1329227995784915871895000743748659792,

%U 226156424291633194186662080095093570015284114833799899656335137245499581360

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

%H G. C. Greubel, <a href="/A323817/b323817.txt">Table of n, a(n) for n = 0..11</a>

%F Logarithmic transform of A323816.

%e The a(3) = 12 set-systems:

%e   {{1, 2, 3}}

%e   {{1, 2}, {1, 3}}

%e   {{1, 2}, {2, 3}}

%e   {{1, 3}, {2, 3}}

%e   {{1, 2}, {1, 2, 3}}

%e   {{1, 3}, {1, 2, 3}}

%e   {{2, 3}, {1, 2, 3}}

%e   {{1, 2}, {1, 3}, {2, 3}}

%e   {{1, 2}, {1, 3}, {1, 2, 3}}

%e   {{1, 2}, {2, 3}, {1, 2, 3}}

%e   {{1, 3}, {2, 3}, {1, 2, 3}}

%e   {{1, 2}, {1, 3}, {2, 3},{1, 2, 3}}

%e The A323816(4) - a(4) = 3 disconnected set-systems covering n vertices with no singletons:

%e   {{1, 2}, {3, 4}}

%e   {{1, 3}, {2, 4}}

%e   {{1, 4}, {2, 3}}

%p b:= n-> add(2^(2^(n-j)-n+j-1)*binomial(n, j)*(-1)^j, j=0..n):

%p a:= proc(n) option remember; b(n)-`if`(n=0, 0, add(

%p        k*binomial(n, k)*b(n-k)*a(k), k=1..n-1)/n)

%p     end:

%p seq(a(n), n=0..8);  # _Alois P. Heinz_, Jan 30 2019

%t nn=10;

%t ser=Sum[Sum[(-1)^(n-k)*Binomial[n,k]*2^(2^k-k-1),{k,0,n}]*x^n/n!,{n,0,nn}];

%t Table[SeriesCoefficient[1+Log[ser],{x,0,n}]*n!,{n,0,nn}]

%o (Magma)

%o m:=10;

%o A323816:= func< n | (&+[(-1)^(n-j)*Binomial(n,j)*2^(2^j -j-1): j in [0..n]]) >;

%o f:= func< x | 1 + Log((&+[A323816(j)*x^j/Factorial(j): j in [0..m+2]])) >;

%o R<x>:=PowerSeriesRing(Rationals(), m+1);

%o Coefficients(R!(Laplace( f(x) ))); // _G. C. Greubel_, Oct 05 2022

%o (SageMath)

%o m=10

%o def A323816(n): return sum((-1)^j*binomial(n,j)*2^(2^(n-j) -n+j-1) for j in range(n+1))

%o def A323817_list(prec):

%o     P.<x> = PowerSeriesRing(QQ, prec)

%o     return P( 1 + log(sum(A323816(j)*x^j/factorial(j) for j in range(m+2))) ).egf_to_ogf().list()

%o A323817_list(m) # _G. C. Greubel_, Oct 05 2022

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

%K nonn

%O 0,4

%A _Gus Wiseman_, Jan 30 2019