A129502 -id:A129502 - cp's OEIS frontend

A317673 Moebius transform of A129502.

1, 2, -1, 3, -1, -2, -1, 4, 0, -2, -1, -3, -1, -2, 1, 5, -1, 0, -1, -3, 1, -2, -1, -4, 0, -2, 0, -3, -1, 2, -1, 6, 1, -2, 1, 0, -1, -2, 1, -4, -1, 2, -1, -3, 0, -2, -1, -5, 0, 0, 1, -3, -1, 0, 1, -4, 1, -2, -1, 3, -1, -2, 0, 7, 1, 2, -1, -3, 1, 2, -1, 0, -1

Offset: 1

Andrew Howroyd, Aug 03 2018

Dirichlet convolution of A209635 and A209229.

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

First column of A129501.

Cf. A129502, A209229, A209635 (Moebius transform of A104117).

a[n_] := Module[{e}, Sum[e = IntegerExponent[d, 2]; If[d == 2^e, MoebiusMu[n/d] Binomial[2 + e, 2], 0], {d, Divisors[n]}]];
a /@ Range[1, 100] (* Jean-François Alcover, Sep 24 2019, from PARI *)
f[p_, e_] := If[e == 1, -1, 0]; f[2, e_] := e+1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 70] (* Amiram Eldar, Aug 28 2023 *)

a(n)={sumdiv(n, d,  my(e=valuation(d, 2)); if(d==1<

Multiplicative with a(2^e) = e+1, and if p is an odd prime, a(p) = -1 and a(p^e) = 0 for e >= 2. - Amiram Eldar, Aug 28 2023

A129501 A103994 * A115361.

Original entry on oeis.org

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

A129503 Pascal's Fredholm-Rueppel triangle.

Original entry on oeis.org

1, 1, 1, 1, 2, 0, 1, 3, 0, 1, 1, 4, 0, 3, 0, 1, 5, 0, 6, 0, 0, 1, 6, 0, 10, 0, 0, 0, 1, 7, 0, 15, 0, 0, 0, 1, 1, 8, 0, 21, 0, 0, 0, 4, 0, 1, 9, 0, 28, 0, 0, 0, 10, 0, 0, 1, 10, 0, 36, 0, 0, 0, 20, 0, 0, 0, 1, 11, 0, 45, 0, 0, 0, 35, 0, 0, 0, 0, 1, 12, 0, 55, 0, 0, 0, 56, 0, 0, 0, 0, 0

Offset: 1

Gary W. Adamson, Apr 18 2007

First row of the array = the Fredholm-Rueppel sequence (A036987); which becomes the right border of the triangle. Second row of the array (1, 2, 0, 3, 0, 0, 0, 4, ...) = A104117. Third row of the array (1, 3, 0, 6, 0, 0, 0, 10, ...) = A129502. Row sums of triangle A129503 = A129504: (1, 2, 3, 5, 8, 12, 17, 24, 34, ...).

			First few rows of the triangle:
  1;
  1,  1;
  1,  2,  0;
  1,  3,  0,  1;
  1,  4,  0,  3,  0;
  1,  5,  0,  6,  0,  0;
  1,  6,  0, 10,  0,  0,  0;
  1,  7,  0, 15,  0,  0,  0,  1;
  1,  8,  0, 21,  0,  0,  0,  4,  0;
  1,  9,  0, 28,  0,  0,  0, 10,  0,  0;
  1, 10,  0, 36,  0,  0,  0, 20,  0,  0,  0;
  ...

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

Row sums are A129504.

Cf. A036987, A115361, A104117, A129502.

T(n,k)=my(e=valuation(k,2)); if(k==2^e, binomial(n-k+e, e)) \\ Andrew Howroyd, Aug 09 2018

Antidiagonals of an array in which n-th row (n=0,1,2,...) = M^n * V, where M = A115361 as an infinite lower triangular matrix and V = the Fredholm-Rueppel sequence A036987 as a vector: [1, 1, 0, 1, 0, 0, 0, 1, ...]. The array = 1, 1, 0, 1, 0, 0, 0, 1, 0, ... 1, 2, 0, 3, 0, 0, 0, 4, 0, ... 1, 3, 0, 6, 0, 0, 0, 10, 0, ... 1, 4, 0, 10, 0, 0, 0, 20, 0, ... (n+1)-th row can be generated from A115361 * n-th row.

T(n, 2^e) = binomial(n + e - 2^e, e), T(n, k) = 0 otherwise. - Andrew Howroyd, Aug 09 2018

a(53) corrected and terms a(67) and beyond from Andrew Howroyd, Aug 09 2018

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A317673 Moebius transform of A129502.

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A129501 A103994 * A115361.

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A129503 Pascal's Fredholm-Rueppel triangle.

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