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A168386 Arithmetic derivative of the double factorial of n.

%I A168386 #20 Jun 13 2022 13:01:33
%S A168386 0,0,1,1,12,8,112,71,1472,1269,17408,14904,270336,204147,4199424,
%T A168386 4143285,87834624,72462870,1797783552,1411253955,40414740480,
%U A168386 36183623805,937430876160,845972658090,26095323709440,24311657884500,707908274749440,869872809558375
%N A168386 Arithmetic derivative of the double factorial of n.
%H A168386 Alois P. Heinz, <a href="/A168386/b168386.txt">Table of n, a(n) for n = 0..800</a>
%F A168386 a(n) = A003415(A006882(n)). - _R. J. Mathar_, Nov 26 2009
%p A168386 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:
%p A168386 A168386 := proc(n) A003415(doublefactorial(n)) ; end proc:
%p A168386 seq(A168386(n),n=0..80) ; # _R. J. Mathar_, Nov 26 2009
%p A168386 # second Maple program:
%p A168386 d:= n-> n*add(i[2]/i[1], i=ifactors(n)[2]):
%p A168386 a:= proc(n) option remember;
%p A168386       `if`(n<2, 0, a(n-2)*n+doublefactorial(n-2)*d(n))
%p A168386     end:
%p A168386 seq(a(n), n=0..40);  # _Alois P. Heinz_, Jun 06 2015
%t A168386 d[n_] := n*Total[#2/#1& @@@ FactorInteger[n]];
%t A168386 a[0] = a[1] = 0; a[n_] := d[n!!];
%t A168386 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, May 18 2018 *)
%o A168386 (Python 3.8+)
%o A168386 from collections import Counter
%o A168386 from sympy import factorial2, factorint
%o A168386 def A168386(n): return sum((factorial2(n)*e//p for p,e in sum((Counter(factorint(m)) for m in range(n,1,-2)),start=Counter({2:0})).items())) if n > 1 else 0 # _Chai Wah Wu_, Jun 12 2022
%Y A168386 Cf. A003415, A068311.
%K A168386 easy,nonn
%O A168386 0,5
%A A168386 _Paolo P. Lava_ and _Giorgio Balzarotti_, Nov 24 2009
%E A168386 Program replaced by a structured program - _R. J. Mathar_, Nov 26 2009