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 1-5 of 5 results.

A338684 a(n) = Sum_{d|n} (-1)^(d-1) * (n/d)^n * binomial(d+n/d-1, d).

Original entry on oeis.org

1, 7, 82, 975, 15626, 275817, 5764802, 133561087, 3486981232, 99853521768, 3138428376722, 106947820494048, 3937376385699290, 155549105311903523, 6568409424129452048, 295137771929866797055, 14063084452067724991010, 708228596784096039676230, 37589973457545958193355602
Offset: 1

Views

Author

Seiichi Manyama, Apr 23 2021

Keywords

Crossrefs

Cf. A338663, A338682, A338683, A338685, A338689.

Programs

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

A340626 a(n) = Sum_{d|n, d odd} binomial(d+n/d-1, d).

Original entry on oeis.org

1, 2, 4, 4, 6, 10, 8, 8, 20, 16, 12, 32, 14, 22, 72, 16, 18, 84, 20, 76, 142, 34, 24, 144, 152, 40, 248, 148, 30, 518, 32, 32, 398, 52, 828, 620, 38, 58, 600, 832, 42, 1416, 44, 408, 2864, 70, 48, 864, 1766, 2078, 1192, 612, 54, 3224, 4424, 3488, 1598, 88, 60, 6784, 62, 94, 13528, 64, 8634
Offset: 1

Views

Author

Seiichi Manyama, Apr 25 2021

Keywords

Links

Crossrefs

Cf. A081543, A338682, A340625.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, Binomial[# + n/# - 1, #] &, OddQ[#] &]; Array[a, 65] (* Amiram Eldar, Apr 25 2021 *)
  • PARI
    a(n) = sumdiv(n, d, (d%2)*binomial(d+n/d-1, d));
  • PARI
    N=66; x='x+O('x^N); Vec(sum(k=1, N, 1/(1-x^k)^k-1/(1+x^k)^k)/2)
  • PARI
    N=66; x='x+O('x^N); Vec(sum(k=1, N, x^(2*k-1)/(1-x^(2*k-1))^(2*k)))

Formula

G.f.: (1/2) * Sum_{k >= 1} (1/(1 - x^k)^k - 1/(1 + x^k)^k).
G.f.: Sum_{k >= 1} x^(2*k-1)/(1 - x^(2*k-1))^(2*k).
a(n) = (A081543(n) + A338682(n))/2.
If p is prime, a(p) = (p mod 2) + p.

A344777 a(n) = Sum_{d|n} (-1)^(n/d-1) * binomial(d+n/d-1, d).

Original entry on oeis.org

1, -1, 4, -6, 6, -3, 8, -22, 20, 0, 12, -44, 14, 7, 72, -95, 18, -10, 20, -71, 142, 33, 24, -399, 152, 52, 248, -57, 30, -121, 32, -679, 398, 102, 828, -685, 38, 133, 600, -1568, 42, -140, 44, 318, 2864, 207, 48, -5858, 1766, -751, 1192, 831, 54, 348, 4424, -3979, 1598, 348, 60
Offset: 1

Views

Author

Seiichi Manyama, May 28 2021

Keywords

Links

Crossrefs

Cf. A081543, A217670, A338682.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (-1)^(n/# - 1) * Binomial[# + n/# - 1, #] &]; Array[a, 60] (* Amiram Eldar, May 28 2021 *)
  • PARI
    a(n) = sumdiv(n, d, (-1)^(n/d-1)*binomial(d+n/d-1, d));
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1+x^k)^(k+1)))

Formula

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

A357041 a(n) = Sum_{d|n} 2^(d-1) * binomial(d+n/d-1,d).

Original entry on oeis.org

1, 4, 7, 18, 21, 66, 71, 196, 305, 648, 1035, 2526, 4109, 8774, 16875, 34288, 65553, 134860, 262163, 531506, 1051237, 2109594, 4194327, 8425348, 16779257, 33611984, 67123631, 134350206, 268435485, 537178750, 1073741855, 2148064768, 4295048345, 8591114580
Offset: 1

Views

Author

Seiichi Manyama, Feb 26 2023

Keywords

Crossrefs

Cf. A081543, A338682.
Cf. A360797.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, 2^(#-1) * Binomial[# + n/# - 1, #] &]; Array[a, 50] (* Amiram Eldar, Jul 31 2023 *)
  • PARI
    a(n) = sumdiv(n, d, 2^(d-1)*binomial(d+n/d-1, d));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (1/(1-2*x^k)^k-1))/2)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (2*x)^k/(1-x^k)^(k+1))/2)
  • Python
    from math import comb
from sympy import divisors
def A357041(n): return sum(comb(d+n//d-1,d)<Chai Wah Wu, Feb 27 2023

Formula

G.f.: (1/2) * Sum_{k>0} (1/(1 - 2 * x^k)^k - 1).
G.f.: (1/2) * Sum_{k>0} (2 * x)^k/(1 - x^k)^(k+1).
If p is prime, a(p) = p + 2^(p-1).
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.