A129501 -id:A129501 - cp's OEIS frontend

A129360 A054525 * A115361.

1, 0, 1, -1, 0, 1, 0, 0, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 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 10 2007

Row sums = A209229 (1, 1, 0, 1, 0, 0, 0, 1, ...).

A129353 = the inverse Möbius transform of A115361.

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

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

Column 1 is A087003 (Moebius transform of A209229).
Row sums are A209229.
Cf. A054525, A115361, A129353, A001511, A103994, A129501.

tabl(nn) = {Tm = matrix(nn, nn, n, k, if (! (n % k), moebius(n/k), 0)); Tr = matrix(nn, nn, n, k, n--; k--; if ((n==k), 1, if (n==2*k+1, -1, 0))); Ti = Tr^(-1); Tp = Tm*Ti; for (n=1, nn, for (k=1, n, print1(Tp[n, k], ", ");); print(););} \\ Michel Marcus, Mar 28 2015

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

Moebius transform of A115361.

T(n,k) = A087003(n/k) for k | n, T(n,k) = 0 otherwise. - Andrew Howroyd, Aug 03 2018

More terms from Michel Marcus, Mar 28 2015

Offset changed by 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

A317673 Moebius transform of A129502.

Original entry on oeis.org

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

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A129360 A054525 * A115361.

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

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A317673 Moebius transform of A129502.

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