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

%I #17 Jun 11 2018 08:02:41

%S 1,1,4,16,70,280,1168,4672,18884,75632,303368,1213472,4858064,

%T 19432256,77743040,310975520,1243959873,4975839492,19903598208,

%U 79614392832,318458493192,1273834028832,5095339755744,20381359022976,81525450936496,326101803775384

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

%H Alois P. Heinz, <a href="/A144316/b144316.txt">Table of n, a(n) for n = 1..500</a>

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

%F a(n) ~ c * 4^n, where c = 0.0724091505138381672774074945426621544789572745186499358668403190389... . - _Vaclav Kotesovec_, Sep 03 2014

%p k:=2: 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, 2];

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

%Y 2nd column of A144324, 3rd column of A144823.

%K eigen,nonn

%O 1,3

%A _Alois P. Heinz_, Sep 17 2008