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

1, 1, 3, 9, 30, 90, 288, 864, 2647, 7968, 24084, 72252, 217467, 652401, 1958931, 5877333, 17637453, 52912359, 158754606, 476263818, 1428840972, 4286528100, 12859728804, 38579186412, 115738013592, 347214043476, 1041643435230, 3124930353363, 9374794990911

Offset: 1

Author

Alois P. Heinz, Sep 21 2008

Keywords

• eigen
• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..700
• N. J. A. Sloane, Transforms

Crossrefs

2nd column of A144823.

Programs

Maple

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);

Mathematica

dc[b_, c_] := Module[{f}, f[n_] := f[n] = Sum[b[d] c[n/d], {d, If[n<0, {}, Divisors[n]]}]; f];
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]];
a[n_] := A[n, 2];
Array[a, 30] (* Jean-François Alcover, Dec 18 2020, after Maple *)

Formula

a(n) ~ c * 3^n, where c = 0.1365983596534181021630692308337960543393478528568767041107748567859... . - Vaclav Kotesovec, Sep 03 2014