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.

Previous Showing 11-14 of 14 results.

A363667 a(n) = Sum_{d|n} (n/d)^(n-n/d) * binomial(d+n-2,n-1).

Original entry on oeis.org

1, 3, 7, 37, 71, 751, 925, 13161, 45676, 262911, 184757, 18014557, 2704157, 133062875, 2838201061, 16907954129, 601080391, 830283170617, 9075135301, 87074953375981, 246003195539410, 53321730394923, 2104098963721, 479275771000215865, 1952680410445479976
Offset: 1

Views

Author

Seiichi Manyama, Jun 14 2023

Keywords

Crossrefs

Cf. A342628, A359112, A363650.
Cf. A363664, A363666.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^(n-n/#) * Binomial[# + n - 2, n - 1] &]; Array[a, 25] (* Amiram Eldar, Jul 12 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^(n-n/d)*binomial(d+n-2, n-1));

Formula

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

A342828 a(n) = Sum_{d|n} (-1)^(n/d+1) * d^(n-d).

Original entry on oeis.org

1, 0, 2, -4, 2, -11, 2, -320, 731, -2869, 2, -1827, 2, -819447, 10297068, -33570816, 2, 1775078476, 2, -36222872973, 678610493340, -285310622035, 2, 169888943418701, 95367431640627, -302875089815037, 150094917726535604, -569376395999240231, 2, 104002456598734754865, 2
Offset: 1

Views

Author

Seiichi Manyama, Mar 23 2021

Keywords

Links

Crossrefs

Cf. A308367, A321438, A342628.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (-1)^(n/# + 1) * #^(n - #) &]; Array[a, 30] (* Amiram Eldar, Mar 23 2021 *)
  • PARI
    a(n) = sumdiv(n, d, (-1)^(n/d+1)*d^(n-d));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1+(k*x)^k)))

Formula

G.f.: Sum_{k>=1} x^k/(1 + (k * x)^k).
If p is an odd prime, a(p) = 2.

A359206 a(n) = Sum_{d|n} 4^(n-d).

Original entry on oeis.org

1, 5, 17, 81, 257, 1345, 4097, 20737, 69633, 328705, 1048577, 5574657, 16777217, 83902465, 286261249, 1359020033, 4294967297, 22565617665, 68719476737, 348967141377, 1168499539969, 5497562333185, 17592186044417, 93531519582209, 282574488338433
Offset: 1

Views

Author

Seiichi Manyama, Dec 20 2022

Keywords

Crossrefs

Cf. A074854, A112329, A357051.
Cf. A342628, A342629, A359204.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, 4^(n-#) &]; Array[a, 25] (* Amiram Eldar, Aug 23 2023 *)
  • PARI
    a(n) = sumdiv(n, d, 4^(n-d));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, 4^(k-1)*x^k/(1-4^(k-1)*x^k)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-(4*x)^k)))

Formula

G.f.: Sum_{k>=1} 4^(k-1) * x^k/(1 - 4^(k-1) * x^k).
G.f.: Sum_{k>=1} x^k/(1 - (4 * x)^k).

A359442 a(n) = Sum_{d|n} d^(n + 1 - d - n/d).

Original entry on oeis.org

1, 2, 2, 4, 2, 15, 2, 74, 83, 643, 2, 12635, 2, 117715, 397188, 2359426, 2, 103572204, 2, 1260918355, 13841818644, 25937425627, 2, 5612318393211, 152587890627, 23298085126579, 1853020231898564, 2422197090649523, 2, 1032944452284531101, 2, 10376297939508166658
Offset: 1

Views

Author

Seiichi Manyama, Jan 14 2023

Keywords

Links

Crossrefs

Cf. A294645, A342628, A342629, A342677, A359700.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^(n + 1 - # - n/#) &]; Array[a, 32] (* Amiram Eldar, Aug 09 2023 *)
  • PARI
    a(n) = sumdiv(n, d, d^(n+1-d-n/d));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-(k*x)^k/k)))

Formula

G.f.: Sum_{k>0} x^k / (1 - (k * x)^k / k).
If p is prime, a(p) = 2.
Previous Showing 11-14 of 14 results.

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