A168386 Arithmetic derivative of the double factorial of n.

0, 0, 1, 1, 12, 8, 112, 71, 1472, 1269, 17408, 14904, 270336, 204147, 4199424, 4143285, 87834624, 72462870, 1797783552, 1411253955, 40414740480, 36183623805, 937430876160, 845972658090, 26095323709440, 24311657884500, 707908274749440, 869872809558375

Offset: 0

Author

Paolo P. Lava and Giorgio Balzarotti, Nov 24 2009

Keywords

• easy
• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..800

Crossrefs

Cf. A003415, A068311.

Programs

Maple

A003415 := proc(n) local pfs ; if n <= 1 then 0 ; else pfs := ifactors(n)[2] ; n*add(op(2,p)/op(1,p),p=pfs) ; fi; end proc:
A168386 := proc(n) A003415(doublefactorial(n)) ; end proc:
seq(A168386(n),n=0..80) ; # R. J. Mathar, Nov 26 2009
# second Maple program:
d:= n-> n*add(i[2]/i[1], i=ifactors(n)[2]):
a:= proc(n) option remember;
      `if`(n<2, 0, a(n-2)*n+doublefactorial(n-2)*d(n))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jun 06 2015

Mathematica

d[n_] := n*Total[#2/#1& @@@ FactorInteger[n]];
a[0] = a[1] = 0; a[n_] := d[n!!];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, May 18 2018 *)

Formula

a(n) = A003415(A006882(n)). - R. J. Mathar, Nov 26 2009

Extensions

Program replaced by a structured program - R. J. Mathar, Nov 26 2009