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A259409
The pi-based arithmetic derivative of the double factorial of n.
2
0, 0, 1, 2, 12, 19, 128, 193, 1600, 2997, 20224, 37692, 319488, 552366, 5164032, 10853055, 103268352, 198691110, 2199453696, 4050806490, 49934499840, 102089892240, 1176592711680, 2471811316695, 32489204613120, 71282307214125, 893769083781120, 2351538388135125
OFFSET
0,4
LINKS
FORMULA
a(n) = A258851(n!!) = A258851(A006882(n)).
MAPLE
with(numtheory):
d:= n-> n*add(i[2]*pi(i[1])/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);
MATHEMATICA
d[n_] := n*Sum[i[[2]]*PrimePi[i[[1]]]/i[[1]], {i, FactorInteger[n]}];
a[n_] := a[n] = If[n < 2, 0, a[n-2]*n + (n-2)!!*d[n]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, May 02 2022, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jun 26 2015
STATUS
approved