A190186 Numerator of expression W_n occurring in analysis of bubble sort.

1, 2, 10, 29, 97, 739, 6331, 8617, 633127, 1037497, 90414391, 1214394319, 17506484887, 38519714137, 4419404086711, 10972377997177, 1410921315134167, 27316952872520239, 555986170009834231, 154130283599461067, 265123004099257677847, 883735015159907270617, 150492959376114678237751, 293138621437723505079883, 100289605416287509517021527

Offset: 1

Author

N. J. A. Sloane, May 05 2011

Keywords

• nonn
• frac

Examples

			1, 2, 10/3, 29/6, 97/15, 739/90, 6331/630, 8617/720, 633127/45360, 1037497/64800, ...
		

References

• D. E. Knuth, The Art of Computer Programming, Vol. 3, Section 5.2.2, p. 129.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..445

Crossrefs

Cf. A190187.

Programs

Maple

W:=proc(n) local t1,r,s;
t1:=add( add(s!*r^(n-s), s=r+1..n), r=0..n-1);
t1/n!;
end;

Mathematica

Numerator[Table[n! + Sum[ Sum[s!*k^(n - s), {s, k + 1, n}], {k, 1, n - 1}]/n!, {n, 1, 50}]] (* G. C. Greubel, Dec 29 2017 *)

PARI

for(n=1,30, print1(numerator(1 + sum(k=1,n-1, sum(s=k+1, n, s!*k^(n-s)))/n!), ", ")) \\ G. C. Greubel, Dec 29 2017

Formula

W_n = Sum_{r=0..(n-1)}( Sum_{s=(r+1)..n} s!*r^(n-s) )/n!.

W_n = numerator(A190194(n)/n!).