A190194 a(n) = Sum_{r=0..n-1} Sum_{s=r+1..n} s! * r^(n-s).

0, 1, 4, 20, 116, 776, 5912, 50648, 482552, 5065016, 58099832, 723315128, 9715154552, 140051879096, 2157103991672, 35355232693688, 614453167841912, 11287370521073336, 218535622980161912, 4447889360078673848, 94944254697268017272, 2120984032794061422776, 49489160848954807154552, 1203943675008917425902008, 30486416629523244528307832

Offset: 0

Author

N. J. A. Sloane, May 05 2011

Keywords

• nonn

Comments

The expression a(n)/n! = A190186/A190187 arises in the analysis of bubble sort [Knuth].

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 = 0..445

Crossrefs

Cf. A190186, A190187.

Programs

Maple

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

Mathematica

Join[{0}, Table[Sum[n! + Sum[s!*k^(n - s), {s, k + 1, n - 1}], {k, 0, n - 1}], {n, 1, 50}]] (* G. C. Greubel, Dec 28 2017 *)

PARI

for(n=0,30, print1(if(n==0,0, sum(k=0,n-1, n! + sum(s=k+1,n-1, s!*k^(n-s)))), ", ")) \\ G. C. Greubel, Dec 28 2017