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A078631 Denominators of coefficients of asymptotic expansion of probability p(n) (see A002816) in powers of 1/n.

%I A078631 #10 Apr 10 2012 14:26:42
%S A078631 1,1,1,3,3,15,45,63,63,405,14175,51975,93555,15795,42567525,49116375,
%T A078631 91216125,2170943775,19538493975,109185701625,3093594879375,
%U A078631 10257709336875,428772250281375,281764621613475,158210081654625,160789593855515625
%N A078631 Denominators of coefficients of asymptotic expansion of probability p(n) (see A002816) in powers of 1/n.
%H A078631 Vaclav Kotesovec, <a href="/A078631/b078631.txt">Table of n, a(n) for n = 0..50</a>
%H A078631 B. Aspvall and F. M. Liang, <a href="http://www-db.stanford.edu/TR/cstr8x.html">The dinner table problem</a>, Technical Report CS-TR-80-829, Computer Science Department, Stanford, California, 1980.
%e A078631 p(n) = exp(-2)*(1 - 4/n + 20/(3n^3) + 58/(3n^4) + ...).
%t A078631 t = 15;
%t A078631 y[n_]:=(1+Sum[Subscript[p,k]/n^k,{k,1,t}]);
%t A078631 mul=1;start=9; If[t>9,mul=n^(t-9);start=t];
%t A078631 w=Apart[Expand[mul*Simplify[
%t A078631 y[n]*n*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*(n-7)*(n-8)*(n-9)*(n-10)
%t A078631 -((3*n-30)*y[n-11]
%t A078631 +(6*n-45)*y[n-10]*(n-10)
%t A078631 +(5*n+18)*y[n-9]*(n-9)*(n-10)
%t A078631 -(8*n-139)*y[n-8]*(n-8)*(n-9)*(n-10)
%t A078631 -(26*n-204)*y[n-7]*(n-7)*(n-8)*(n-9)*(n-10)
%t A078631 -(4*n-30)*y[n-6]*(n-6)*(n-7)*(n-8)*(n-9)*(n-10)
%t A078631 +(26*n-148)*y[n-5]*(n-5)*(n-6)*(n-7)*(n-8)*(n-9)*(n-10)
%t A078631 +(8*n-74)*y[n-4]*(n-4)*(n-5)*(n-6)*(n-7)*(n-8)*(n-9)*(n-10)
%t A078631 -(9*n-18)*y[n-3]*(n-3)*(n-4)*(n-5)*(n-6)*(n-7)*(n-8)*(n-9)*(n-10)
%t A078631 -(2*n-15)*y[n-2]*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*(n-7)*(n-8)*(n-9)*(n-10)
%t A078631 +(n+2)*y[n-1]*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*(n-7)*(n-8)*(n-9)*(n-10))],n],n];
%t A078631 sol=Solve[Table[Coefficient[w,n,j]==0,{j,start,start-t+1,-1}]];
%t A078631 asympt=y[n]/.sol[[1]];
%t A078631 Table[Denominator[Coefficient[asympt,n,-j]],{j,0,t}] (* _Vaclav Kotesovec_, Apr 06 2012 *)
%Y A078631 Cf. A078630, A089222.
%K A078631 nonn
%O A078631 0,4
%A A078631 _N. J. A. Sloane_, Dec 13 2002
%E A078631 Terms a(8)-a(25) from Vaclav Kotesovec, Apr 06 2012