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Shifts left when Dirichlet convolution (DC:(b,b)->a) applied 3 times.
3

%I #11 Jun 11 2018 08:02:33

%S 1,1,8,64,540,4320,35008,280064,2244152,17955008,143670304,1149362432,

%T 9195171392,73561371136,588492929536,4707943678208,37663565234758,

%U 301308521878064,2410468302643136,19283746421145088,154269972376667232

%N Shifts left when Dirichlet convolution (DC:(b,b)->a) applied 3 times.

%H Alois P. Heinz, <a href="/A144317/b144317.txt">Table of n, a(n) for n = 1..500</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%p k:=3: with (numtheory): dc:= proc(b,c) proc(n) option remember; add (b(d) *c(n/d), d=`if`(n<0,{}, divisors(n))) end end: a:='a': b[1]:= dc(a,a): for t from 2 to k do b[t]:= dc(b[t-1], b[t-1]) od: a:= n-> `if`(n=1, 1, b[k](n-1)): seq (a(n), n=1..30);

%t dc[b_, c_] := Module[{proc}, proc[n_] := proc[n] = Sum[b[d]*c[n/d], {d, If[n < 0, {}, Divisors[n]]}]; proc];

%t A[n_, k_] := Module[{a, b, t}, b[1] = dc[a, a]; For[t = 2, t <= k, t++, b[t] = dc[b[t-1], b[t-1]]]; a = Function[m, If[m == 1, 1, b[k][m - 1]]]; a[n]];

%t a[n_] := A[n, 3];

%t Array[a, 30] (* _Jean-François Alcover_, Jun 11 2018, after _Alois P. Heinz_ *)

%Y 3rd column of A144324, 7th column of A144823.

%K eigen,nonn

%O 1,3

%A _Alois P. Heinz_, Sep 17 2008