A259409 The pi-based arithmetic derivative of the double factorial of n.

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

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

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

Cf. A006882, A258845, A258851.

Sequence in context: A109299 A216629 A337290 * A073257 A240905 A303880

Adjacent sequences: A259406 A259407 A259408 * A259410 A259411 A259412

Keyword

nonn

Author

Alois P. Heinz, Jun 26 2015

Status

approved