cp's OEIS Frontend

A274273 Number of noncomposite areas of a Venn diagram for n multisets.

%I A274273 #45 Mar 07 2025 02:53:45
%S A274273 1,2,8,50,392,3602,37928,451250,5995592,88073042,1418137448,
%T A274273 24846302450,470675213192,9587626273682,209000505036968,
%U A274273 4855088300025650,119739457665173192,3124793129198573522,86030517992814720488,2492084621605727380850,75769449406015305475592
%N A274273 Number of noncomposite areas of a Venn diagram for n multisets.
%C A274273 Noncomposite areas are the smallest areas in the figures, those that are not composed of smaller areas.
%C A274273 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.
%C A274273 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.
%H A274273 Aurelian Radoaca, <a href="https://drive.google.com/file/d/1g9_ZCgnSKOibsoKeQUhH807OG6Bfw0eP/view">Properties of Multisets Compared to Sets</a>, unpublished article, 2023.
%F A274273 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.
%F A274273 a(n) ~ n!*n / (2*log(2)^(n+2)). - _Vaclav Kotesovec_, Jul 04 2016
%F A274273 From _Peter Bala_, May 21 2017: (Start)
%F A274273 a(n) = 1 - 2*A000670(n) + A000670(n+1) for n >= 1.
%F A274273 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)
%F A274273 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
%p A274273 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
%t A274273 F[0] = 1; F[n_] := F[n] = Sum[Binomial[n, k] F[n - k], {k, 1, n}];
%t A274273 a[0] := 1; a[n_] := 1 + F[n] + 2 Sum[Binomial[n, i] F[i] F[n - i], {i, 1, n - 1}];
%t A274273 Table[a[n], {n, 0, 20}]
%Y A274273 Cf. A000670, A343583.
%K A274273 nonn,easy
%O A274273 0,2
%A A274273 _Aurelian Radoaca_, Jun 17 2016