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

1, -4, 0, 20, 58, 796, 7858, 40324, 140194, 2444744, 40680494, -7117319032, -149539443124, -223750776484, -4960419494993024, -46146161037854692, -689434674121075448, -132496988938839119444, -9686633414582239854958, -442788087926096759821484

Offset: 0

Author

N. J. A. Sloane, Dec 13 2002

Keywords

• sign

Examples

			p(n) = exp(-2)*(1 - 4/n + 20/(3n^3) + 58/(3n^4) + ...).
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..50
• B. Aspvall and F. M. Liang, The dinner table problem, Technical Report CS-TR-80-829, Computer Science Department, Stanford, California, 1980.

Crossrefs

Cf. A078631, A089222.

Programs

Mathematica

t = 15;
y[n_]:=(1+Sum[Subscript[p,k]/n^k,{k,1,t}]);
mul=1;start=9; If[t>9,mul=n^(t-9);start=t];
w=Apart[Expand[mul*Simplify[
y[n]*n*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*(n-7)*(n-8)*(n-9)*(n-10)
-((3*n-30)*y[n-11]
+(6*n-45)*y[n-10]*(n-10)
+(5*n+18)*y[n-9]*(n-9)*(n-10)
-(8*n-139)*y[n-8]*(n-8)*(n-9)*(n-10)
-(26*n-204)*y[n-7]*(n-7)*(n-8)*(n-9)*(n-10)
-(4*n-30)*y[n-6]*(n-6)*(n-7)*(n-8)*(n-9)*(n-10)
+(26*n-148)*y[n-5]*(n-5)*(n-6)*(n-7)*(n-8)*(n-9)*(n-10)
+(8*n-74)*y[n-4]*(n-4)*(n-5)*(n-6)*(n-7)*(n-8)*(n-9)*(n-10)
-(9*n-18)*y[n-3]*(n-3)*(n-4)*(n-5)*(n-6)*(n-7)*(n-8)*(n-9)*(n-10)
-(2*n-15)*y[n-2]*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*(n-7)*(n-8)*(n-9)*(n-10)
+(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];
sol=Solve[Table[Coefficient[w,n,j]==0,{j,start,start-t+1,-1}]];
asympt=y[n]/.sol[[1]];
Table[Numerator[Coefficient[asympt,n,-j]],{j,0,t}] (* Vaclav Kotesovec, Apr 06 2012 *)

Extensions

Terms a(5)-a(19) from Vaclav Kotesovec, Apr 06 2012 (terms a(5)-a(7) were wrong, see A089222 for more information)