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 1-7 of 7 results.

A332470 a(n) = Sum_{d|n} mu(n/d) * binomial(n+d-2, n-1).

Original entry on oeis.org

1, 1, 5, 16, 69, 226, 923, 3312, 12825, 47896, 184755, 700712, 2704155, 10373455, 40113421, 154946976, 601080389, 2332498482, 9075135299, 35338355380, 137846298360, 538213522254, 2104098963719, 8233142596640, 32247603662625, 126408753954731, 495918514791900
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 13 2020

Keywords

Links

Crossrefs

Cf. A000010, A007438, A008683, A117108, A117109, A332508, A343547.

Programs

  • Magma
    [&+[MoebiusMu(n div d) *Binomial(n+d-2,n-1):d in Divisors(n)]:n in [1..30]]; // Marius A. Burtea, Feb 13 2020
  • Mathematica
    Table[DivisorSum[n, MoebiusMu[n/#] Binomial[n + # - 2, n - 1] &], {n, 1, 27}]
Table[SeriesCoefficient[Sum[MoebiusMu[k] x^k/(1 - x^k)^n, {k, 1, n}], {x, 0, n}], {n, 1, 27}]
  • PARI
    a(n) = sumdiv(n, d, moebius(n/d)*binomial(n+d-2, n-1)); \\ Michel Marcus, Feb 14 2020

Formula

a(n) = [x^n] Sum_{k>=1} mu(k) * x^k / (1 - x^k)^n.
a(n) = |{(x_1, x_2, ... , x_{n-1}) : 1 <= x_1 <= x_2 <= ... <= x_n = n, gcd(x_1, x_2, ... , x_n) = 1}|. - Seiichi Manyama, Apr 20 2021

A343544 a(n) = n * Sum_{d|n} binomial(d+2,3)/d.

Original entry on oeis.org

1, 6, 13, 32, 40, 94, 91, 184, 204, 320, 297, 612, 468, 770, 850, 1184, 986, 1752, 1349, 2280, 2114, 2662, 2323, 4184, 3125, 4264, 4266, 5740, 4524, 7660, 5487, 8352, 7546, 9180, 8470, 13212, 9176, 12654, 12194, 16640, 12382, 19628, 14233, 20724, 19590, 22034, 18471, 30416, 21462
Offset: 1

Views

Author

Seiichi Manyama, Apr 19 2021

Keywords

Links

Crossrefs

Cf. A000203, A038040, A059358, A309731, A343545, A343546, A343547.

Programs

  • Maple
    f:= n -> n/6*add((d+1)*(d+2),d=numtheory:-divisors(n)):
map(f, [$1..100]); # Robert Israel, Apr 26 2021
  • Mathematica
    a[n_] := n * DivisorSum[n, Binomial[# + 2, 3]/# &]; Array[a, 50] (* Amiram Eldar, Apr 25 2021 *)
  • PARI
    a(n) = n*sumdiv(n, d, binomial(d+2, 3)/d);
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, binomial(k+2, 3)*x^k/(1-x^k)^2))

Formula

G.f.: Sum_{k>=1} k * x^k/(1 - x^k)^4 = Sum_{k>=1} binomial(k+2,3) * x^k/(1 - x^k)^2.

A343549 a(n) = n * Sum_{d|n} binomial(d+n-1,n)/d.

Original entry on oeis.org

1, 5, 13, 49, 131, 545, 1723, 6809, 24484, 94445, 352727, 1366273, 5200313, 20135939, 77571083, 301034537, 1166803127, 4540794476, 17672631919, 68943346009, 269129827042, 1052178506615, 4116715363823, 16124644677569, 63205303337656, 247964681424725, 973469783435197
Offset: 1

Views

Author

Seiichi Manyama, Apr 19 2021

Keywords

Links

Crossrefs

Cf. A000203, A038040, A309731, A343544, A343545, A343546, A343547, A343548.

Programs

  • Mathematica
    a[n_] := n * DivisorSum[n, Binomial[# + n - 1, n]/# &]; Array[a, 30] (* Amiram Eldar, Apr 25 2021 *)
  • PARI
    a(n) = n*sumdiv(n, d, binomial(d+n-1, n)/d);

Formula

a(n) = [x^n] Sum_{k>=1} k * x^k/(1 - x^k)^(n+1).
a(n) = [x^n] Sum_{k>=1} binomial(k+n-1,n) * x^k/(1 - x^k)^2.
From Seiichi Manyama, Jun 14 2023: (Start)
a(n) = Sum_{d|n} binomial(d+n-1,d).
a(n) = [x^n] Sum_{k>=1} (1/(1 - x^k)^n - 1). (End)

A343545 a(n) = n * Sum_{d|n} binomial(d+3,4)/d.

Original entry on oeis.org

1, 7, 18, 49, 75, 177, 217, 428, 549, 890, 1012, 1824, 1833, 2849, 3360, 4732, 4862, 7506, 7334, 10810, 11382, 14729, 14973, 22188, 20850, 27482, 29052, 37408, 35989, 50490, 46407, 61824, 62106, 75854, 75390, 101673, 91427, 116033, 117624, 146680, 135792, 179886, 163228, 208208
Offset: 1

Views

Author

Seiichi Manyama, Apr 19 2021

Keywords

Crossrefs

Cf. A000203, A038040, A073570, A309731, A343544, A343546, A343547.

Programs

  • Mathematica
    a[n_] := n * DivisorSum[n, Binomial[# + 3, 4]/# &]; Array[a, 50] (* Amiram Eldar, Apr 25 2021 *)
  • PARI
    a(n) = n*sumdiv(n, d, binomial(d+3, 4)/d);
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, binomial(k+3, 4)*x^k/(1-x^k)^2))

Formula

G.f.: Sum_{k>=1} k * x^k/(1 - x^k)^5 = Sum_{k>=1} binomial(k+3,4) * x^k/(1 - x^k)^2.

A343546 a(n) = n * Sum_{d|n} binomial(d+4,5)/d.

Original entry on oeis.org

1, 8, 24, 72, 131, 318, 469, 936, 1359, 2294, 3014, 5172, 6201, 9548, 12126, 17376, 20366, 29862, 33668, 47372, 54684, 71874, 80753, 111000, 119410, 154986, 173988, 220864, 237365, 309864, 324663, 411744, 445170, 542776, 578984, 731340, 749435, 918118, 981474
Offset: 1

Views

Author

Seiichi Manyama, Apr 19 2021

Keywords

Crossrefs

Cf. A000203, A038040, A101289, A309731, A343544, A343545, A343547.

Programs

  • Mathematica
    a[n_] := n * DivisorSum[n, Binomial[# + 4, 5]/# &]; Array[a, 40] (* Amiram Eldar, Apr 25 2021 *)
  • PARI
    a(n) = n*sumdiv(n, d, binomial(d+4, 5)/d);
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, binomial(k+4, 5)*x^k/(1-x^k)^2))

Formula

G.f.: Sum_{k>=1} k * x^k/(1 - x^k)^6 = Sum_{k>=1} binomial(k+4,5) * x^k/(1 - x^k)^2.

A343553 a(n) = Sum_{1 <= x_1 <= x_2 <= ... <= x_n = n} gcd(x_1, x_2, ... , x_n).

Original entry on oeis.org

1, 3, 8, 26, 74, 287, 930, 3572, 12966, 49379, 184766, 710712, 2704168, 10427822, 40123208, 155289768, 601080406, 2334740919, 9075135318, 35352194658, 137846990678, 538302226835, 2104098963742, 8233721100024, 32247603765020, 126412458921072, 495918569262798
Offset: 1

Views

Author

Seiichi Manyama, Apr 19 2021

Keywords

Examples

			a(3) = gcd(1,1,3) + gcd(1,2,3) + gcd(1,3,3) + gcd(2,2,3) + gcd(2,3,3) + gcd(3,3,3) = 1 + 1 + 1 + 1 + 1 + 3 = 8.

Links

Crossrefs

Cf. A000010, A332470, A332508, A343516, A343517, A343547.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, EulerPhi[n/#] * Binomial[# + n - 2, n-1] &]; Array[a, 30] (* Amiram Eldar, Apr 25 2021 *)
  • PARI
    a(n) = sumdiv(n, d, eulerphi(n/d)*binomial(d+n-2, n-1));

Formula

a(n) = A343516(n,n-1).
a(n) = Sum_{d|n} phi(n/d) * binomial(d+n-2, n-1).
a(n) = [x^n] Sum_{k >= 1} phi(k) * x^k/(1 - x^k)^n.
a(n) ~ 2^(2*n - 2) / sqrt(Pi*n). - Vaclav Kotesovec, May 23 2021

A343567 a(n) = Sum_{d|n} (n/d)^(n/d) * binomial(d+n-2,n-1).

Original entry on oeis.org

1, 6, 33, 292, 3195, 47154, 824467, 16783176, 387434574, 10000082730, 285311855367, 8916101760828, 302875109296409, 11112006847596746, 437893890421433595, 18446744074133995664, 827240261886937844567, 39346408075305857765940
Offset: 1

Views

Author

Seiichi Manyama, Apr 20 2021

Keywords

Crossrefs

Cf. A062796, A343547, A343568.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^(n/#) * Binomial[# + n - 2, n - 1] &]; Array[a, 20] (* Amiram Eldar, Apr 25 2021 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^(n/d)*binomial(d+n-2, n-1));

Formula

a(n) = [x^n] Sum_{k>=1} (k * x)^k/(1 - x^k)^n.
Showing 1-7 of 7 results.

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