A274273 Number of noncomposite areas of a Venn diagram for n multisets.
1, 2, 8, 50, 392, 3602, 37928, 451250, 5995592, 88073042, 1418137448, 24846302450, 470675213192, 9587626273682, 209000505036968, 4855088300025650, 119739457665173192, 3124793129198573522, 86030517992814720488, 2492084621605727380850, 75769449406015305475592
Offset: 0
Links
- Aurelian Radoaca, Properties of Multisets Compared to Sets, unpublished article, 2023.
Programs
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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
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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}]
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
Comments