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.

A338685 a(n) = Sum_{d|n} d^n * binomial(d, n/d).

Original entry on oeis.org

1, 8, 81, 1040, 15625, 282123, 5764801, 134610944, 3486804084, 100097656250, 3138428376721, 107025924222976, 3937376385699289, 155582338242342053, 6568408660888671875, 295155786482995691520, 14063084452067724991009, 708240750793407501694308, 37589973457545958193355601
Offset: 1

Views

Author

Seiichi Manyama, Apr 23 2021

Keywords

Links

Crossrefs

Cf. A023887, A318636, A327238, A338684, A338693.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^n * Binomial[#, n/#] &]; Array[a, 20] (* Amiram Eldar, Apr 24 2021 *)
  • PARI
    a(n) = sumdiv(n, d, d^n*binomial(d, n/d));
  • PARI
    N=20; x='x+O('x^N); Vec(sum(k=1, N, (1+(k*x)^k)^k-1))

Formula

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

A327124 Expansion of Sum_{k>=1} ((1 - (-x)^k)^k - 1).

Original entry on oeis.org

1, -2, 3, -3, 5, -3, 7, -2, 10, 0, 11, -1, 13, 7, 25, 13, 17, -2, 19, 30, 56, 33, 23, 1, 26, 52, 111, 98, 29, -51, 31, 158, 198, 102, 56, 24, 37, 133, 325, 304, 41, -189, 43, 517, 626, 207, 47, 191, 50, -2, 731, 988, 53, -435, 517, 1315, 1026, 348, 59, 18
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 14 2019

Keywords

Links

Crossrefs

Cf. A056045, A112329, A318636, A327238.

Programs

  • Mathematica
    nmax = 60; CoefficientList[Series[Sum[((1 - (-x)^k)^k - 1), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, (-1)^(n - #) Binomial[n/#, #] &], {n, 1, 60}]
  • PARI
    a(n)={sumdiv(n, d, (-1)^(n-d) * binomial(n/d,d))} \\ Andrew Howroyd, Sep 14 2019

Formula

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

A327249 Expansion of Sum_{k>=1} x^k * (1 + k * x^k)^k.

Original entry on oeis.org

1, 2, 1, 5, 1, 14, 1, 17, 28, 26, 1, 160, 1, 50, 251, 321, 1, 622, 1, 1607, 1030, 122, 1, 6257, 3126, 170, 2917, 12202, 1, 27291, 1, 28929, 6656, 290, 84036, 117721, 1, 362, 13183, 407121, 1, 417881, 1, 220100, 850312, 530, 1, 2246465, 823544, 2100626
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 15 2019

Keywords

Crossrefs

Cf. A006005 (positions of 1's), A087909, A217668, A260180, A327238.

Programs

  • Magma
    [&+[(n div d)^(d-1)*Binomial(n div d,d-1):d in Divisors(n)]:n in [1..50]]; // Marius A. Burtea, Sep 15 2019
  • Mathematica
    nmax = 50; CoefficientList[Series[Sum[x^k (1 + k x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, (n/#)^(# - 1) Binomial[n/#, # - 1] &], {n, 1, 50}]
  • PARI
    a(n) = sumdiv(n, d, (n/d)^(d-1) * binomial(n/d,d-1)); \\ Michel Marcus, Sep 15 2019

Formula

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

A338683 a(n) = - Sum_{d|n} (-n/d)^d * binomial(d+n/d-1, d).

Original entry on oeis.org

1, 3, 10, 3, 26, 13, 50, -177, 352, -84, 122, -996, 170, -153, 9704, -13313, 290, -6518, 362, -2771, 107986, 17073, 530, -805070, 394376, 99984, 1203580, 1196313, 842, -4500745, 962, -13313025, 14199222, 2316234, 33547310, -19898071, 1370, 10418613, 168405072
Offset: 1

Views

Author

Seiichi Manyama, Apr 23 2021

Keywords

Crossrefs

Cf. A327238, A338662, A338682, A338684, A338688.

Programs

  • Mathematica
    a[n_] := -DivisorSum[n, (-n/#)^# * Binomial[# + n/# - 1, #] &]; Array[a, 40] (* Amiram Eldar, Apr 24 2021 *)
  • PARI
    a(n) = -sumdiv(n, d, (-n/d)^d*binomial(d+n/d-1, d));
  • PARI
    N=66; x='x+O('x^N); Vec(sum(k=1, N, 1-1/(1+k*x^k)^k))

Formula

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

A376016 a(n) = Sum_{d|n} d^(n/d) * binomial(n/d,d).

Original entry on oeis.org

1, 2, 3, 8, 5, 30, 7, 104, 36, 330, 11, 1296, 13, 2702, 2445, 7440, 17, 33030, 19, 51220, 76566, 112662, 23, 699216, 3150, 639002, 1653399, 2064412, 29, 10620300, 31, 12451872, 29229288, 17825826, 1640660, 190101888, 37, 89653286, 455976417, 441305440, 41
Offset: 1

Views

Author

Seiichi Manyama, Sep 06 2024

Keywords

Links

Crossrefs

Cf. A055225, A327238, A376020.

Programs

  • PARI
    a(n) = sumdiv(n, d, d^(n/d)*binomial(n/d, d));
  • PARI
    my(N=50, x='x+O('x^N)); Vec(sum(k=1, N, (k*x^k)^k/(1-k*x^k)^(k+1)))
  • Python
    from math import comb
from itertools import takewhile
from sympy import divisors
def A376016(n): return sum(d**(m:=n//d)*comb(m,d) for d in takewhile(lambda d:d**2<=n,divisors(n))) # Chai Wah Wu, Sep 06 2024

Formula

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

A360759 a(n) = Sum_{d|n} d^(d+n/d) * binomial(d,n/d).

Original entry on oeis.org

1, 16, 243, 4112, 78125, 1680345, 40353607, 1073766400, 31381060338, 1000000781250, 34522712143931, 1283918489808640, 51185893014090757, 2177953338656796883, 98526125335697265625, 4722366482899710050304, 239072435685151324847153
Offset: 1

Views

Author

Seiichi Manyama, Feb 19 2023

Keywords

Links

Crossrefs

Cf. A318636, A327238, A338685, A338693, A338694.
Cf. A360712.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^(# + n/#) * Binomial[#, n/#] &]; Array[a, 20] (* Amiram Eldar, Aug 02 2023 *)
  • PARI
    a(n) = sumdiv(n, d, d^(d+n/d)*binomial(d, n/d));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, k^k*((1+k*x^k)^k-1)))

Formula

G.f.: Sum_{k>0} k^k * ( (1 + k*x^k)^k - 1 ).
If p is prime, a(p) = p^(p+2).
Showing 1-6 of 6 results.

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