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The pi-based arithmetic derivative of the double factorial of n.
2

%I #9 May 02 2022 11:09:13

%S 0,0,1,2,12,19,128,193,1600,2997,20224,37692,319488,552366,5164032,

%T 10853055,103268352,198691110,2199453696,4050806490,49934499840,

%U 102089892240,1176592711680,2471811316695,32489204613120,71282307214125,893769083781120,2351538388135125

%N The pi-based arithmetic derivative of the double factorial of n.

%H Alois P. Heinz, <a href="/A259409/b259409.txt">Table of n, a(n) for n = 0..800</a>

%F a(n) = A258851(n!!) = A258851(A006882(n)).

%p with(numtheory):

%p d:= n-> n*add(i[2]*pi(i[1])/i[1], i=ifactors(n)[2]):

%p a:= proc(n) option remember;

%p `if`(n<2, 0, a(n-2)*n+doublefactorial(n-2)*d(n))

%p end:

%p seq(a(n), n=0..40);

%t d[n_] := n*Sum[i[[2]]*PrimePi[i[[1]]]/i[[1]], {i, FactorInteger[n]}];

%t a[n_] := a[n] = If[n < 2, 0, a[n-2]*n + (n-2)!!*d[n]];

%t Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, May 02 2022, after _Alois P. Heinz_ *)

%Y Cf. A006882, A258845, A258851.

%K nonn

%O 0,4

%A _Alois P. Heinz_, Jun 26 2015