A144366 Shifts 2 places left under Dirichlet convolution.

1, 1, 1, 2, 2, 5, 4, 12, 8, 28, 17, 60, 34, 134, 68, 276, 140, 580, 280, 1186, 560, 2436, 1128, 4906, 2256, 9976, 4516, 20020, 9048, 40324, 18096, 80860, 36192, 162320, 72418, 324920, 144852, 651177, 289704, 1302914, 579476, 2608360, 1158952

Offset: 1

Author

Alois P. Heinz, Sep 18 2008

Keywords

• eigen
• nonn

Comments

Appears to be essentially the same as a sequence in Section 2.2 of Baril-Ramirez. - N. J. A. Sloane, Feb 18 2024

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• Jean-Luc Baril and José L. Ramírez, Knight's paths towards Catalan numbers, Univ. Bourgogne Franche-Comté (2022). Also arXiv:2206.12087, Jan 2023.
• N. J. A. Sloane, Transforms

Crossrefs

2nd column of A144374. Cf. A000005.

Programs

Maple

k:= 2: with(numtheory): dck:= proc(b,c) proc(n, k) option remember; add(b(d,k) *c(n/d,k), d=`if`(n<0,{}, divisors(n))) end end: B:= dck(T,T): T:= (n, k)-> if n<=k then 1 else B(n-k, k) fi: a:= n-> T(n,k): seq(a(n), n=1..50);

Mathematica

k = 2; dck[b_, c_][n_, k_] := dck[b, c][n, k] = DivisorSum[n, b[#, k] * c[n/#, k]&]; B = dck[T, T]; T[n_, k_] :=  If[n <= k, 1, B[n-k, k]]; a[n_] := T[n, k]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Apr 05 2017, translated from Maple *)

Formula

G.f.: x + x^2 * (1 + Sum_{i>=1} Sum_{j>=1} a(i)*a(j)*x^(i*j)). - Ilya Gutkovskiy, May 09 2019