cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A338657 a(n) = Sum_{d|n} mu(d) * binomial(d+n/d-1, d).

Original entry on oeis.org

1, 1, 2, 1, 4, -3, 6, -2, -1, -10, 10, -22, 12, -21, -40, -20, 16, -55, 18, -80, -98, -55, 22, -90, -101, -78, -138, -182, 28, -271, 30, -104, -330, -136, -756, -37, 36, -171, -520, -676, 40, -476, 42, -550, -1786, -253, 46, 648, -1667, -1276, -1088, -832, 52, 1539, -4312
Offset: 1

Views

Author

Seiichi Manyama, Apr 22 2021

Keywords

Links

Crossrefs

Cf. A008683, A338656, A343565, A338658.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, MoebiusMu[#] * Binomial[# + n/# - 1, #] &]; Array[a, 100] (* Amiram Eldar, Apr 22 2021 *)
  • PARI
    a(n) = sumdiv(n, d, moebius(d)*binomial(d+n/d-1, d));
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, moebius(k)*x^k/(1-x^k)^(k+1)))

Formula

G.f.: Sum_{k >= 1} mu(k) * x^k/(1 - x^k)^(k+1).
If p is prime, a(p) = p - 1.

A360823 Expansion of Sum_{k>0} k * x^k / (1 - k * x^k)^(k+1).

Original entry on oeis.org

1, 4, 6, 20, 10, 96, 14, 256, 288, 650, 22, 4200, 26, 4004, 11160, 18784, 34, 70758, 38, 164140, 196098, 136664, 46, 1756728, 393800, 747890, 3287844, 5452076, 58, 22563060, 62, 31220032, 50767926, 20059286, 41640130, 391194396, 74, 99622016, 725647728, 1298396440
Offset: 1

Views

Author

Seiichi Manyama, Feb 22 2023

Keywords

Crossrefs

Cf. A338658, A360794, A360824.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^(n/#) * Binomial[# + n/# - 1, #] &]; Array[a, 40] (* Amiram Eldar, Jul 31 2023 *)
  • PARI
    my(N=50, x='x+O('x^N)); Vec(sum(k=1, N, k*x^k/(1-k*x^k)^(k+1)))
  • PARI
    a(n) = sumdiv(n, d, d^(n/d)*binomial(d+n/d-1, d));

Formula

a(n) = Sum_{d|n} d^(n/d) * binomial(d+n/d-1,d).
If p is prime, a(p) = 2 * p.

A338655 a(n) = Sum_{d|n} phi(d) * binomial(d+n/d-1, d).

Original entry on oeis.org

1, 3, 5, 9, 9, 22, 13, 32, 35, 53, 21, 121, 25, 96, 177, 166, 33, 297, 37, 491, 417, 218, 45, 1002, 549, 297, 705, 1375, 57, 2418, 61, 1640, 1405, 491, 3887, 4659, 73, 606, 2233, 8156, 81, 8989, 85, 6189, 11955, 872, 93, 16550, 10387, 12927, 4757, 11111, 105, 22392, 25757
Offset: 1

Views

Author

Seiichi Manyama, Apr 22 2021

Keywords

Crossrefs

Cf. A000010, A156834, A343517, A338658.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, EulerPhi[#] * Binomial[# + n/# - 1, #] &]; Array[a, 100] (* Amiram Eldar, Apr 22 2021 *)
  • PARI
    a(n) = sumdiv(n, d, eulerphi(d)*binomial(d+n/d-1, d));
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k/(1-x^k)^(k+1)))

Formula

G.f.: Sum_{k >= 1} phi(k) * x^k/(1 - x^k)^(k+1).
If p is prime, a(p) = 2*p - 1.
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = Sum_{k=1..n} binomial(gcd(n,k) + n/gcd(n,k) - 1,n/gcd(n,k)).
a(n) = Sum_{k=1..n} binomial(gcd(n,k) + n/gcd(n,k) - 1,gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)
Showing 1-3 of 3 results.

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