OFFSET
0,2
COMMENTS
Noncomposite areas are the smallest areas in the figures, those that are not composed of smaller areas.
As in the case of sets, we consider a universal multiset U and an area external to all multisets represented in the Venn diagram, the difference between U and the union of the multisets.
The difference between the total number of noncomposite areas and the number of disjoint areas in a Venn diagram for n multisets is given by (1 + F(n) + 2*Sum_{i=1..n-1} C(n,i)*F(i)*F(n-i)) - (1 + F(n) + Sum_{i=1..n-1} C(n,i)*F(i)) = Sum_{i=1..n-1} C(n,i)*F(i)*(2*F(n-i)-1), where F(n) is A000670.
LINKS
Aurelian Radoaca, Properties of Multisets Compared to Sets, unpublished article, 2023.
FORMULA
a(n) = 1 + F(n) + 2*Sum_{i=1..n-1} C(n,i)*F(i)*F(n-i) for n > 1, where a(0)=1, a(1)=2, and F(i) is A000670.
a(n) ~ n!*n / (2*log(2)^(n+2)). - Vaclav Kotesovec, Jul 04 2016
From Peter Bala, May 21 2017: (Start)
G.f.: A(x) = 1 + 2*x/(1 - x)*( 1 + 3*x/(1 - 3*x)*( 1 + 4*x/(1 - 4*x)*( 1 + 5*x/(1 - 5*x)*( 1 + .... (End)
a(n) = 1 + (1/2)*Li_{-n-1}(1/2) - Li_{-n}(1/2) = A343583(n) + 1, where Li_{n}x) is the polylogarithm function. - Peter Luschny, Apr 26 2021
EXAMPLE
a(0)=1, a(1)=2.
MAPLE
seq(1 + add(factorial(k)*(stirling2(n+1, k) - 2*stirling2(n, k)), k = 0..n+1), n = 1..20); # Peter Bala, May 21 2017
MATHEMATICA
F[0] = 1; F[n_] := F[n] = Sum[Binomial[n, k] F[n - k], {k, 1, n}];
a[0] := 1; a[n_] := 1 + F[n] + 2 Sum[Binomial[n, i] F[i] F[n - i], {i, 1, n - 1}];
Table[a[n], {n, 0, 20}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Aurelian Radoaca, Jun 17 2016
STATUS
approved