A144316 Shifts left when Dirichlet convolution (DC:(b,b)->a) applied twice.

1, 1, 4, 16, 70, 280, 1168, 4672, 18884, 75632, 303368, 1213472, 4858064, 19432256, 77743040, 310975520, 1243959873, 4975839492, 19903598208, 79614392832, 318458493192, 1273834028832, 5095339755744, 20381359022976, 81525450936496, 326101803775384

Offset: 1

Author

Alois P. Heinz, Sep 17 2008

Keywords

• eigen
• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..500
• N. J. A. Sloane, Transforms

Crossrefs

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

Programs

Maple

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);

Mathematica

dc[b_, c_] := Module[{proc}, proc[n_] := proc[n] = Sum[b[d]*c[n/d], {d, If[n < 0, {}, Divisors[n]]}]; proc];
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]];
a[n_] := A[n, 2];
Array[a, 30] (* Jean-François Alcover, Jun 11 2018, after Alois P. Heinz *)

Formula

a(n) ~ c * 4^n, where c = 0.0724091505138381672774074945426621544789572745186499358668403190389... . - Vaclav Kotesovec, Sep 03 2014