This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A259409 #9 May 02 2022 11:09:13
%S A259409 0,0,1,2,12,19,128,193,1600,2997,20224,37692,319488,552366,5164032,
%T A259409 10853055,103268352,198691110,2199453696,4050806490,49934499840,
%U A259409 102089892240,1176592711680,2471811316695,32489204613120,71282307214125,893769083781120,2351538388135125
%N A259409 The pi-based arithmetic derivative of the double factorial of n.
%H A259409 Alois P. Heinz, <a href="/A259409/b259409.txt">Table of n, a(n) for n = 0..800</a>
%F A259409 a(n) = A258851(n!!) = A258851(A006882(n)).
%p A259409 with(numtheory):
%p A259409 d:= n-> n*add(i[2]*pi(i[1])/i[1], i=ifactors(n)[2]):
%p A259409 a:= proc(n) option remember;
%p A259409 `if`(n<2, 0, a(n-2)*n+doublefactorial(n-2)*d(n))
%p A259409 end:
%p A259409 seq(a(n), n=0..40);
%t A259409 d[n_] := n*Sum[i[[2]]*PrimePi[i[[1]]]/i[[1]], {i, FactorInteger[n]}];
%t A259409 a[n_] := a[n] = If[n < 2, 0, a[n-2]*n + (n-2)!!*d[n]];
%t A259409 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, May 02 2022, after _Alois P. Heinz_ *)
%Y A259409 Cf. A006882, A258845, A258851.
%K A259409 nonn
%O A259409 0,4
%A A259409 _Alois P. Heinz_, Jun 26 2015