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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 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:=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. = 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

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