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

1,2

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

Formula

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

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A190187.

Sequence in context: A295525 A302274 A047112 * A032250 A215790 A133485

Adjacent sequences: A190183 A190184 A190185 * A190187 A190188 A190189

Keyword

nonn,frac

Author

N. J. A. Sloane, May 05 2011

Status

approved