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 1-6 of 6 results.

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)

A363663 a(n) = Sum_{d|n} (n/d)^(d-1) * binomial(d+n-1,n).

Original entry on oeis.org

1, 4, 11, 46, 127, 596, 1717, 7792, 24806, 108450, 352717, 1563914, 5200301, 22539046, 77876117, 331982444, 1166803111, 4945693769, 17672631901, 74053888812, 269344740908, 1118110015874, 4116715363801, 16984153623296, 63205318063252, 259049084680612
Offset: 1

Views

Author

Seiichi Manyama, Jun 14 2023

Keywords

Crossrefs

Cf. A167531, A363642, A363645.
Cf. A343548, A363661, A363664.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^(#-1) * Binomial[# + n - 1, n] &]; Array[a, 25] (* Amiram Eldar, Jul 12 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^(d-1)*binomial(d+n-1, n));

Formula

a(n) = [x^n] Sum_{k>0} x^k/(1 - k*x^k)^(n+1).

A343565 a(n) = |{(x_1, x_2, ... , x_n) : 1 <= x_1 <= x_2 <= ... <= x_n <= n, gcd(x_1, x_2, ... , x_n, n) = 1}|.

Original entry on oeis.org

1, 2, 9, 30, 125, 428, 1715, 6270, 24255, 91367, 352715, 1345448, 5200299, 20019526, 77554749, 300295038, 1166803109, 4535971916, 17672631899, 68913247655, 269128640958, 1051984969598, 4116715363799, 16123381989000, 63205303195125, 247956558998878, 973469689288236
Offset: 1

Views

Author

Seiichi Manyama, Apr 20 2021

Keywords

Links

Crossrefs

Cf. A000010, A007438, A117108, A117109, A332470, A343517, A343548, A343549.

Programs

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

Formula

a(n) = Sum_{d|n} mu(n/d) * binomial(d+n-1, n).
a(n) = [x^n] Sum_{k>=1} mu(k) * x^k/(1 - x^k)^(n+1).

A363660 a(n) = Sum_{d|n} binomial(d+n,n).

Original entry on oeis.org

2, 9, 24, 90, 258, 1043, 3440, 13419, 48850, 187836, 705444, 2725099, 10400614, 40233015, 155133856, 601820876, 2333606238, 9079958260, 35345263820, 137876637843, 538259060526, 2104292500739, 8233430727624, 32248866496625, 126410606580284
Offset: 1

Views

Author

Seiichi Manyama, Jun 14 2023

Keywords

Crossrefs

Cf. A007503, A116963, A363628.
Cf. A000984, A134758, A343548, A343549, A363661.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, Binomial[# + n, n] &]; Array[a, 25] (* Amiram Eldar, Jul 17 2023 *)
  • PARI
    a(n) = sumdiv(n, d, binomial(d+n, n));

Formula

a(n) = [x^n] Sum_{k>0} (1/(1 - x^k)^(n+1) - 1).
a(n) = [x^n] Sum_{k>0} binomial(k+n,n) * x^k/(1 - x^k).

A363664 a(n) = Sum_{d|n} (n/d)^(n-n/d) * binomial(d+n-1,n).

Original entry on oeis.org

1, 4, 11, 56, 127, 1100, 1717, 19300, 64406, 383010, 352717, 23214660, 5200301, 191172406, 3465549077, 20859527460, 1166803111, 1010698826825, 17672631901, 102589250081802, 286539905316908, 75260204476154, 4116715363801, 548610025890719156
Offset: 1

Views

Author

Seiichi Manyama, Jun 14 2023

Keywords

Crossrefs

Cf. A342628, A359112, A363650.
Cf. A343548, A363662, A363663, A363667.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^(n-n/#) * Binomial[# + n - 1, n] &]; Array[a, 25] (* Amiram Eldar, Jul 12 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^(n-n/d)*binomial(d+n-1, n));

Formula

a(n) = [x^n] Sum_{k>0} x^k/(1 - (k*x)^k)^(n+1).

A366986 Square array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{d|n} binomial(d+k-1,k).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 4, 4, 3, 1, 5, 7, 7, 2, 1, 6, 11, 14, 6, 4, 1, 7, 16, 25, 16, 12, 2, 1, 8, 22, 41, 36, 31, 8, 4, 1, 9, 29, 63, 71, 71, 29, 15, 3, 1, 10, 37, 92, 127, 147, 85, 50, 13, 4, 1, 11, 46, 129, 211, 280, 211, 145, 52, 18, 2, 1, 12, 56, 175, 331, 498, 463, 371, 176, 74, 12, 6
Offset: 1

Views

Author

Seiichi Manyama, Oct 31 2023

Keywords

Examples

			Square  array begins:
  1,  1,  1,  1,   1,   1,   1, ...
  2,  3,  4,  5,   6,   7,   8, ...
  2,  4,  7, 11,  16,  22,  29, ...
  3,  7, 14, 25,  41,  63,  92, ...
  2,  6, 16, 36,  71, 127, 211, ...
  4, 12, 31, 71, 147, 280, 498, ...
  2,  8, 29, 85, 211, 463, 925, ...

Crossrefs

Columns k=0..5 give A000005, A000203, A007437, A059358, A073570, A101289.
T(n,n-1) gives A332508.
T(n,n) gives A343548.
Cf. A366977.

Programs

  • PARI
    T(n, k) = sumdiv(n, d, binomial(d+k-1, k));

Formula

G.f. of column k: Sum_{j>=1} x^j/(1 - x^j)^(k+1).
Showing 1-6 of 6 results.

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