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%I #16 Sep 08 2022 08:45:12
%S 1,1,3,7,41,161,1387,7687,86865,623233,8682131,76586951,1265108473,
%T 13257387937,252846968571,3071345365831,66334014084257,
%U 916952261126657,22098449760227875,342676322992004743,9109114481334332361,156647957565343927201
%N 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.
%D 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
%H G. C. Greubel, <a href="/A089656/b089656.txt">Table of n, a(n) for n = 0..445</a>
%H Mark A. Shattuck and Carl G. Wagner, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/Shattuck2/shattuck44.html">Parity Theorems for Statistics on Lattice Paths and Laguerre Configurations</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.1.
%F 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)).
%F 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
%F a(n) ~ exp(3*n^(1/3)/2 - n) * n^n / sqrt(3). - _Vaclav Kotesovec_, Nov 14 2017
%e 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)
%t 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 *)
%o (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
%o (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]];
%K nonn
%O 0,3
%A Carl G. Wagner (wagner(AT)math.utk.edu), Jan 15 2004
%E More terms from _Vaclav Kotesovec_, Nov 14 2017
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