cp's OEIS Frontend

A078630 Numerators of coefficients of asymptotic expansion of probability p(n) (see A002816) in powers of 1/n.

%I A078630 #10 Apr 10 2012 14:26:11
%S A078630 1,-4,0,20,58,796,7858,40324,140194,2444744,40680494,-7117319032,
%T A078630 -149539443124,-223750776484,-4960419494993024,-46146161037854692,
%U A078630 -689434674121075448,-132496988938839119444,-9686633414582239854958,-442788087926096759821484
%N A078630 Numerators of coefficients of asymptotic expansion of probability p(n) (see A002816) in powers of 1/n.
%H A078630 Vaclav Kotesovec, <a href="/A078630/b078630.txt">Table of n, a(n) for n = 0..50</a>
%H A078630 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 A078630 p(n) = exp(-2)*(1 - 4/n + 20/(3n^3) + 58/(3n^4) + ...).
%t A078630 t = 15;
%t A078630 y[n_]:=(1+Sum[Subscript[p,k]/n^k,{k,1,t}]);
%t A078630 mul=1;start=9; If[t>9,mul=n^(t-9);start=t];
%t A078630 w=Apart[Expand[mul*Simplify[
%t A078630 y[n]*n*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*(n-7)*(n-8)*(n-9)*(n-10)
%t A078630 -((3*n-30)*y[n-11]
%t A078630 +(6*n-45)*y[n-10]*(n-10)
%t A078630 +(5*n+18)*y[n-9]*(n-9)*(n-10)
%t A078630 -(8*n-139)*y[n-8]*(n-8)*(n-9)*(n-10)
%t A078630 -(26*n-204)*y[n-7]*(n-7)*(n-8)*(n-9)*(n-10)
%t A078630 -(4*n-30)*y[n-6]*(n-6)*(n-7)*(n-8)*(n-9)*(n-10)
%t A078630 +(26*n-148)*y[n-5]*(n-5)*(n-6)*(n-7)*(n-8)*(n-9)*(n-10)
%t A078630 +(8*n-74)*y[n-4]*(n-4)*(n-5)*(n-6)*(n-7)*(n-8)*(n-9)*(n-10)
%t A078630 -(9*n-18)*y[n-3]*(n-3)*(n-4)*(n-5)*(n-6)*(n-7)*(n-8)*(n-9)*(n-10)
%t A078630 -(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 A078630 +(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 A078630 sol=Solve[Table[Coefficient[w,n,j]==0,{j,start,start-t+1,-1}]];
%t A078630 asympt=y[n]/.sol[[1]];
%t A078630 Table[Numerator[Coefficient[asympt,n,-j]],{j,0,t}] (* _Vaclav Kotesovec_, Apr 06 2012 *)
%Y A078630 Cf. A078631, A089222.
%K A078630 sign
%O A078630 0,2
%A A078630 _N. J. A. Sloane_, Dec 13 2002
%E A078630 Terms a(5)-a(19) from Vaclav Kotesovec, Apr 06 2012 (terms a(5)-a(7) were wrong, see A089222 for more information)