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A274273
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Number of noncomposite areas of a Venn diagram for n multisets.
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1
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1, 2, 8, 50, 392, 3602, 37928, 451250, 5995592, 88073042, 1418137448, 24846302450, 470675213192, 9587626273682, 209000505036968, 4855088300025650, 119739457665173192, 3124793129198573522, 86030517992814720488, 2492084621605727380850, 75769449406015305475592
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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Table of n, a(n) for n=0..20.
Aurelian Radoaca, Properties of Multisets Compared to Sets, unpublished article, 2016,
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FORMULA
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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)
a(n) = 1 - 2*A000670(n) + A000670(n+1) for n >= 1.
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
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EXAMPLE
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a(0)=1, a(1)=2.
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MAPLE
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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
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MATHEMATICA
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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}]
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CROSSREFS
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Cf. A000670, A343583.
Sequence in context: A360949 A231352 A186182 * A121677 A120956 A000557
Adjacent sequences: A274270 A274271 A274272 * A274274 A274275 A274276
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KEYWORD
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nonn,easy
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AUTHOR
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Aurelian Radoaca, Jun 17 2016
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STATUS
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approved
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