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A089656
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Given a distribution of n balls, labeled 1,...,n, among n unlabeled contents-ordered urns, arrange the nonempty urns in increasing order of their initial elements: U_1,...U_k and sum the quantities (i-1)(card U_i - 1) for i=1,...,k to get the "weight" of this distribution. These numbers represent the number of distributions of even weight minus the number with odd weight.
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1
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1, 1, 3, 7, 41, 161, 1387, 7687, 86865, 623233, 8682131, 76586951, 1265108473, 13257387937, 252846968571, 3071345365831, 66334014084257, 916952261126657, 22098449760227875, 342676322992004743, 9109114481334332361, 156647957565343927201
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OFFSET
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0,3
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REFERENCES
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Mark A. Shattuck and Carl G. Wagner, Parity Theorems for Statistics on Lattice Paths and Laguerre Distributions, Research Report, Mathematics Department, University of Tennessee, Knoxville, TN, 2004
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LINKS
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FORMULA
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E.g.f.: cosh(x*(1-x^2)^(-1/2)) + (1-x^2)^(1/2)*(1-x)^(-1)*sinh(x*(1-x^2)^(-1/2)).
Recurrence: (8*n^2 - 56*n + 61)*a(n) = (8*n^2 - 80*n + 147)*a(n-1) + (24*n^4 - 256*n^3 + 863*n^2 - 1061*n + 258)*a(n-2) - 2*(n-2)*(8*n^3 - 96*n^2 + 305*n - 239)*a(n-3) - (n-3)*(n-2)*(24*n^4 - 320*n^3 + 1431*n^2 - 2295*n + 639)*a(n-4) + (n-5)*(n-4)*(n-3)*(n-2)*(8*n^2 - 64*n + 71)*a(n-5) + (n-6)*(n-5)^2*(n-4)*(n-3)*(n-2)*(8*n^2 - 40*n + 13)*a(n-6). - Vaclav Kotesovec, Nov 14 2017
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EXAMPLE
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a(3)=7 because there are 9 distributions of balls 1,2,3 with weight 0: 123,132,213,231,312,321,12-3,13-2 and 1-2-3 and 2 distributions of weight 1:1-23 and 1-32 (dashes separate contents-ordered urns)
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MATHEMATICA
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nmax = 20; CoefficientList[Series[Cosh[x/Sqrt[1 - x^2]] + Sqrt[1 - x^2] * Sinh[x/Sqrt[1 - x^2]] / (1-x), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 14 2017 *)
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PROG
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(PARI) x='x+O('x^30); Vec(serlaplace(cosh(x*(1-x^2)^(-1/2)) + (1-x^2)^(1/2)*(1-x)^(-1)*sinh(x*(1-x^2)^(-1/2)))) \\ G. C. Greubel, May 23 2018
(Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Cosh(x*(1-x^2)^(-1/2)) + (1-x^2)^(1/2)*(1-x)^(-1)*Sinh(x*(1-x^2)^(-1/2)))); [Factorial(n-1)*b[n]: n in [1..m]];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Carl G. Wagner (wagner(AT)math.utk.edu), Jan 15 2004
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EXTENSIONS
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STATUS
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approved
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