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A258652 Sum of the k-th arithmetic derivative of n-k for k=0..n.

%I A258652 #12 Jun 01 2018 04:04:42
%S A258652 0,1,2,4,5,9,11,16,14,25,36,59,99,209,419,860,1730,3862,9464,21868,
%T A258652 74371,244648,727345,3098351,13469007,56269849,281642632,1406177909,
%U A258652 9597415332,58891421656,411673964638,3406742649805,24202753250241,176482943622608
%N A258652 Sum of the k-th arithmetic derivative of n-k for k=0..n.
%H A258652 Alois P. Heinz, <a href="/A258652/b258652.txt">Table of n, a(n) for n = 0..100</a>
%H A258652 Wikipedia, <a href="https://en.wikipedia.org/wiki/Arithmetic_derivative">Arithmetic derivative</a>
%F A258652 a(n) = Sum_{k=0..n} A258651(n-k,k).
%p A258652 d:= n-> n*add(i[2]/i[1], i=ifactors(n)[2]):
%p A258652 A:= proc(n, k) option remember; `if`(k=0, n, d(A(n, k-1))) end:
%p A258652 a:= proc(n) option remember; add(A(h, n-h), h=0..n) end:
%p A258652 seq(a(n), n=0..40);
%t A258652 d[n_ /; n>1] := n*Sum[i[[2]]/i[[1]], {i, FactorInteger[n]}]; d[_] = 0;
%t A258652 A[n_, k_] := A[n, k] = If[k == 0, n, d[A[n, k-1]]];
%t A258652 a[n_] := a[n] = Sum[A[h, n-h], {h, 0, n}];
%t A258652 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Jun 01 2018, from Maple *)
%Y A258652 Antidiagonal sums of A258651.
%Y A258652 Cf. A003415.
%K A258652 nonn
%O A258652 0,3
%A A258652 _Alois P. Heinz_, Jun 06 2015