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A144817 Shifts left when Dirichlet convolution with a (DC:(b,a)->c) applied twice.

%I A144817 #16 Dec 18 2020 12:01:48
%S A144817 1,1,3,9,30,90,288,864,2647,7968,24084,72252,217467,652401,1958931,
%T A144817 5877333,17637453,52912359,158754606,476263818,1428840972,4286528100,
%U A144817 12859728804,38579186412,115738013592,347214043476,1041643435230,3124930353363,9374794990911
%N A144817 Shifts left when Dirichlet convolution with a (DC:(b,a)->c) applied twice.
%H A144817 Alois P. Heinz, <a href="/A144817/b144817.txt">Table of n, a(n) for n = 1..700</a>
%H A144817 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A144817 a(n) ~ c * 3^n, where c = 0.1365983596534181021630692308337960543393478528568767041107748567859... . - _Vaclav Kotesovec_, Sep 03 2014
%p A144817 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:= proc(n, k) local a, b, t; b[1]:= dc(a,a); for t from 2 to k do b[t]:= dc(b[t-1],a) od: a:= n-> `if`(n=1, 1, b[k](n-1)); a(n) end: a:= n-> A(n,2): seq (a(n), n=1..30);
%t A144817 dc[b_, c_] := Module[{f}, f[n_] := f[n] = Sum[b[d] c[n/d], {d, If[n<0, {}, Divisors[n]]}]; f];
%t A144817 A[n_, k_] := Module[{a, b, t}, b[1] = dc[a, a]; For[t = 2, t <= k, t++, b[t] = dc[b[t-1], a]]; a = Function[m, If[m==1, 1, b[k][m-1]]]; a[n]];
%t A144817 a[n_] := A[n, 2];
%t A144817 Array[a, 30] (* _Jean-François Alcover_, Dec 18 2020, after Maple *)
%Y A144817 2nd column of A144823.
%K A144817 eigen,nonn
%O A144817 1,3
%A A144817 _Alois P. Heinz_, Sep 21 2008