A089656 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.

1, 1, 3, 7, 41, 161, 1387, 7687, 86865, 623233, 8682131, 76586951, 1265108473, 13257387937, 252846968571, 3071345365831, 66334014084257, 916952261126657, 22098449760227875, 342676322992004743, 9109114481334332361, 156647957565343927201

Offset

0,3

References

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..445

Mark A. Shattuck and Carl G. Wagner, Parity Theorems for Statistics on Lattice Paths and Laguerre Configurations, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.1.

Formula

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

a(n) ~ exp(3*n^(1/3)/2 - n) * n^n / sqrt(3). - Vaclav Kotesovec, Nov 14 2017

Example

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)

Mathematica

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 *)

Prog

(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]];

Crossrefs

Sequence in context: A071730 A058815 A333649 * A018970 A018968 A018969

Adjacent sequences: A089653 A089654 A089655 * A089657 A089658 A089659

Keyword

nonn

Author

Carl G. Wagner (wagner(AT)math.utk.edu), Jan 15 2004

Extensions

More terms from Vaclav Kotesovec, Nov 14 2017

Status

approved