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A174586
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Number of n X n (0,1) matrices with two 1's in each row having positive permanent.
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3
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0, 1, 24, 954, 59040, 5295150, 651354480, 105393619800, 21717404916480, 5554438422838200, 1726882980691176000, 641506478978753110800, 280659563041747649760000, 142843312073975729801785200, 83684308104396267184700784000, 55915646244745131440225950320000
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OFFSET
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1,3
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COMMENTS
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a(n) is the normalized volume of the convex hull of (classical) parking functions of length n. - Andrés R. Vindas-Meléndez, Jan 13 2023
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REFERENCES
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Vladimir Shevelev, On the permanent of the stochastic (0,1)-matrices with equal row sums, Izvestia Vuzov of the North-Caucasus region, Nature sciences 1 (1997), 21-38 (in Russian).
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LINKS
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FORMULA
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a(2)=1, for n>=3, a(n) = A001499(n) + Sum_{k=1..n-2} (-1)^(k+1)*k!*(C(n,k))^2*(n-k)^k*a(n-k).
a(n) = n!*((n-1)/2^(n-1))*Sum_{i=0..n-2} (2i+1)!!*C(n-2,i)*(2n-1)^(n-i-2). [corrected by John Lentfer, Oct 05 2022]
For n>=2, a(n) = (n!/2^n)*Sum_{i=0..n} (2i-1)*(2i-1)!!*C(n,i)*(2n-1)^(n-i-1).
a(n) = Gamma(3/4)*(sqrt(2)*Pi*e)^(-1/2)*n!*n^(n-1/4)*(1+O(n^((-1/4)+epsilon) with arbitrary small epsilon>0 for sufficiently large n.
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MATHEMATICA
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Table[n!/2^n * Sum[(2*i-1)*(2*i-1)!!*Binomial[n, i]*(2n-1)^(n-i-1), {i, 0, n}], {n, 1, 20}] (* Vaclav Kotesovec, Nov 30 2017 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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