A317673 -id:A317673 - cp's OEIS frontend

A129501 A103994 * A115361.

1, 2, 1, -1, 0, 1, 3, 2, 0, 1, -1, 0, 0, 0, 1, -2, -1, 2, 0, 0, 1, -1, 0, 0, 0, 0, 0, 1, 4, 3, 0, 2, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 1, -2, -1, 0, 0, 2, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -3, -2, 3, -1, 0, 2, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Apr 17 2007

Row sums = A129502: (1, 3, 0, 6, 0, 0, 0, 10, 0, 0, ...).

			First few rows of the triangle are:
   1;
   2,  1;
  -1,  0,  1;
   3,  2,  0,  1;
  -1,  0,  0,  0,  1;
  -2, -1,  2,  0,  0,  1;
  -1,  0,  0,  0,  0,  0,  1;
   4,  3,  0,  2,  0,  0,  0,  1;
   0,  0, -1,  0,  0,  0,  0,  0,  1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Column 1 is A317673 (Moebius transform of A129502).
Row sums are A129502.
Cf. A103994, A115361, A129360, A000217, A001511.

b[n_] := Module[{e}, Sum[e = IntegerExponent[d, 2]; If[d == 2^e, MoebiusMu[n/d] Binomial[2 + e, 2], 0], {d, Divisors[n]}]];
T[n_, k_] := If[Divisible[n, k], b[n/k], 0];
Table[T[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 24 2019 *)

T(n,k)={ if(n%k, 0, sumdiv(n/k, d, my(e=valuation(d, 2)); if(d==1<Andrew Howroyd, Aug 03 2018

A103994 * A115361 as infinite lower triangular matrices.

T(n,k) = A317673(n/k) for k | n, T(n,k) = 0 otherwise. - Andrew Howroyd, Aug 03 2018

Terms a(56) and beyond from Andrew Howroyd, Aug 03 2018

A129502 For n=2^k, a(n) = binomial(k + 2, 2), else 0.

Original entry on oeis.org

1, 3, 0, 6, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 28, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Gary W. Adamson, Apr 17 2007

Row sums of triangle A129501.

			a(4) = 6 = sum of A129501 terms: (3 + 2 + 0 + 1).

Andrew Howroyd, Table of n, a(n) for n = 1..1024

Row sums of A129501.

Cf. A209229, A104117, A129360, A115361, A103994, A000217, A317673.

Table[If[IntegerQ[Log2[n]],Binomial[Log2[n]+2,2],0],{n,100}] (* Harvey P. Dale, May 10 2022 *)

a(n)={my(e=valuation(n, 2)); if(n==1<Andrew Howroyd, Aug 03 2018

From Andrew Howroyd, Aug 04 2018: (Start)
Multiplicative with a(2^e) = binomial(e + 2, 2), a(p^e) = 0 for odd prime p.
Dirichlet convolution of A104117 and A209229.
a(n) = Sum_{d|n} A104117(n/d) * A209229(d). (End)
Dirichlet g.f.: 1/(1 - 1/2^s)^3. - Amiram Eldar, Oct 28 2023

Name changed and terms a(40) and beyond from Andrew Howroyd, Aug 03 2018

cp's OEIS Frontend

A129501 A103994 * A115361.

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