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 1-10 of 16 results. Next

A327238 Expansion of Sum_{k>=1} ((1 + k * x^k)^k - 1).

Original entry on oeis.org

1, 4, 9, 20, 25, 63, 49, 160, 108, 350, 121, 940, 169, 1225, 1475, 2304, 289, 7560, 361, 8025, 12446, 7139, 529, 58192, 3750, 13858, 61965, 102655, 841, 191181, 961, 318464, 220704, 40460, 354172, 1304370, 1369, 63175, 629863, 4012608, 1681, 1916733, 1849
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 14 2019

Keywords

Links

Crossrefs

Cf. A055225, A056045, A318636, A327124.

Programs

  • Mathematica
    nmax = 43; CoefficientList[Series[Sum[((1 + k x^k)^k - 1), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, (n/#)^# Binomial[n/#, #] &], {n, 1, 43}]
  • PARI
    a(n)={sumdiv(n, d, (n/d)^d * binomial(n/d,d))} \\ Andrew Howroyd, Sep 14 2019

Formula

a(n) = Sum_{d|n} (n/d)^d * binomial(n/d,d).
a(p) = p^2, where p is prime.

A338693 a(n) = Sum_{d|n} d^(d - n/d) * binomial(d, n/d).

Original entry on oeis.org

1, 4, 27, 257, 3125, 46665, 823543, 16777312, 387420490, 10000001250, 285311670611, 8916100467712, 302875106592253, 11112006825910963, 437893890380859625, 18446744073716891649, 827240261886336764177, 39346408075296709766628, 1978419655660313589123979
Offset: 1

Views

Author

Seiichi Manyama, Apr 24 2021

Keywords

Links

Crossrefs

Cf. A318636, A318637, A318638, A338685, A338694.

Programs

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

Formula

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

A318637 Expansion of Sum_{n>=1} ( (2 + x^n)^n - 2^n ).

Original entry on oeis.org

1, 4, 12, 33, 80, 198, 448, 1048, 2305, 5200, 11264, 24824, 53248, 115360, 245800, 526081, 1114112, 2364064, 4980736, 10497290, 22020656, 46165504, 96468992, 201396028, 419430401, 872574976, 1811944704, 3758469400, 7784628224, 16107002892, 33285996544, 68721443936, 141733963008, 292062232576, 601295421524, 1236960724929, 2542620639232, 5222702645248, 10720238663680, 21990282376768
Offset: 1

Views

Author

Paul D. Hanna, Sep 07 2018

Keywords

Examples

			G.f.: A(x) = x + 4*x^2 + 12*x^3 + 33*x^4 + 80*x^5 + 198*x^6 + 448*x^7 + 1048*x^8 + 2305*x^9 + 5200*x^10 + 11264*x^11 + 24824*x^12 + 53248*x^13 + 115360*x^14 + ...
such that
A(x) = x + (2 + x^2)^2 - 2^2 + (2 + x^3)^3 - 2^3 + (2 + x^4)^4 - 2^4 + (2 + x^5)^5 - 2^5 + (2 + x^6)^6 - 2^6 + (2 + x^7)^7 - 2^7 + ...
RELATED SERIES.
The g.f. A(x) equals following series at y = 2:
Sum_{n>=1} ((y + x^n)^n - y^n) = x + 2*y*x^2 + 3*y^2*x^3 + (4*y^3 + 1)*x^4 + 5*y^4*x^5 + (6*y^5 + 3*y)*x^6 + 7*y^6*x^7 + (8*y^7 + 6*y^2)*x^8 + (9*y^8 + 1)*x^9 + (10*y^9 + 10*y^3)*x^10 + 11*y^10*x^11 + (12*y^11 + 15*y^4 + 4*y)*x^12 + 13*y^12*x^13 + (14*y^13 + 21*y^5)*x^14 + (15*y^14 + 10*y^2)*x^15 + (16*y^15 + 28*y^6 + 1)*x^16 + ...

Links

Crossrefs

Cf. A318636, A318638.

Programs

  • PARI
    {a(n) = polcoeff( sum(m=1,n, (x^m + 2 +x*O(x^n))^m - 2^m), n)}
for(n=1,100, print1(a(n),", "))
  • PARI
    a(n) = sumdiv(n, d, 2^(d-n/d)* binomial(d, n/d)); \\ Seiichi Manyama, Apr 24 2021

Formula

a(n) ~ n * 2^(n-1). - Vaclav Kotesovec, Oct 10 2020
a(n) = Sum_{d|n} 2^(d - n/d) * binomial(d, n/d). - Seiichi Manyama, Apr 24 2021
G.f.: Sum_{k >=1} x^(k^2)/(1-2*x^k)^(k+1). - Seiichi Manyama, Oct 30 2023

A338682 a(n) = Sum_{d|n} (-1)^(d-1) * binomial(d+n/d-1, d).

Original entry on oeis.org

1, 1, 4, 0, 6, 3, 8, -8, 20, 0, 12, -12, 14, -7, 72, -65, 18, 10, 20, -61, 142, -33, 24, -203, 152, -52, 248, -183, 30, 121, 32, -617, 398, -102, 828, -619, 38, -133, 600, -896, 42, 140, 44, -870, 2864, -207, 48, -4438, 1766, 751, 1192, -1587, 54, -348, 4424, -3011, 1598, -348, 60
Offset: 1

Views

Author

Seiichi Manyama, Apr 23 2021

Keywords

Links

Crossrefs

Cf. A081543, A217670, A318636, A338683, A338684.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (-1)^(# - 1) * Binomial[# + n/# - 1, #] &]; Array[a, 60] (* Amiram Eldar, Apr 24 2021 *)
  • PARI
    a(n) = sumdiv(n, d, (-1)^(d-1)*binomial(d+n/d-1, d));
  • PARI
    N=66; x='x+O('x^N); Vec(sum(k=1, N, 1-1/(1+x^k)^k))
  • PARI
    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} (1 - 1/(1 + x^k)^k).
G.f.: - Sum_{k >= 1} (-x)^k/(1 - x^k)^(k+1).
If p is prime, a(p) = (-1)^(p-1) + p.

A318638 Expansion of Sum_{n>=1} ( (3 + x^n)^n - 3^n ).

Original entry on oeis.org

1, 6, 27, 109, 405, 1467, 5103, 17550, 59050, 197100, 649539, 2126991, 6908733, 22325625, 71744625, 229602925, 731794257, 2324602206, 7360989291, 23245524600, 73222475256, 230128853031, 721764371007, 2259440202825, 7060738412026, 22029517662984, 68630377426119, 213516777941712, 663426981193869, 2058911488612863, 6382625094934119, 19765549255048254, 61149666233193318
Offset: 1

Views

Author

Paul D. Hanna, Sep 07 2018

Keywords

Examples

			G.f.: A(x) = x + 6*x^2 + 27*x^3 + 109*x^4 + 405*x^5 + 1467*x^6 + 5103*x^7 + 17550*x^8 + 59050*x^9 + 197100*x^10 + 649539*x^11 + 2126991*x^12 + ...
such that
A(x) = x + (3 + x^2)^2 - 3^2 + (3 + x^3)^3 - 3^3 + (3 + x^4)^4 - 3^4 + (3 + x^5)^5 - 3^5 + (3 + x^6)^6 - 3^6 + (3 + x^7)^7 - 3^7 + ...
RELATED SERIES.
The g.f. A(x) equals following series at y = 3:
Sum_{n>=1} ((y + x^n)^n - y^n) = x + 2*y*x^2 + 3*y^2*x^3 + (4*y^3 + 1)*x^4 + 5*y^4*x^5 + (6*y^5 + 3*y)*x^6 + 7*y^6*x^7 + (8*y^7 + 6*y^2)*x^8 + (9*y^8 + 1)*x^9 + (10*y^9 + 10*y^3)*x^10 + 11*y^10*x^11 + (12*y^11 + 15*y^4 + 4*y)*x^12 + 13*y^12*x^13 + (14*y^13 + 21*y^5)*x^14 + (15*y^14 + 10*y^2)*x^15 + (16*y^15 + 28*y^6 + 1)*x^16 + ...

Links

Crossrefs

Cf. A318636, A318637, A338693.

Programs

  • PARI
    {a(n) = polcoeff( sum(m=1,n, (x^m + 3 +x*O(x^n))^m - 3^m), n)}
for(n=1,100, print1(a(n),", "))
  • PARI
    a(n) = sumdiv(n, d, 3^(d-n/d)* binomial(d, n/d)); \\ Seiichi Manyama, Apr 24 2021

Formula

a(n) ~ n * 3^(n-1). - Vaclav Kotesovec, Oct 10 2020
a(n) = Sum_{d|n} 3^(d - n/d) * binomial(d, n/d). - Seiichi Manyama, Apr 24 2021
G.f.: Sum_{k >=1} x^(k^2)/(1-3*x^k)^(k+1). - Seiichi Manyama, Oct 30 2023

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).

A338694 a(n) = Sum_{d|n} d^d * binomial(d, n/d).

Original entry on oeis.org

1, 8, 81, 1028, 15625, 280017, 5764801, 134219264, 3486784428, 100000031250, 3138428376721, 106993206079936, 3937376385699289, 155568095575106627, 6568408355712921875, 295147905179822588160, 14063084452067724991009, 708235345355351624428356, 37589973457545958193355601
Offset: 1

Views

Author

Seiichi Manyama, Apr 24 2021

Keywords

Links

Crossrefs

Cf. A062796, A318636, A318637, A318638, A338693.

Programs

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

Formula

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

A340625 a(n) = Sum_{d|n, d odd, d <= n/d} binomial(n/d, d).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 10, 11, 16, 13, 14, 25, 16, 17, 38, 19, 20, 56, 22, 23, 80, 26, 26, 111, 28, 29, 156, 31, 32, 198, 34, 56, 256, 37, 38, 325, 96, 41, 406, 43, 44, 626, 46, 47, 608, 50, 302, 731, 52, 53, 870, 517, 64, 1026, 58, 59, 1992, 61, 62, 1429, 64, 1352, 1606, 67, 68, 1840, 2192, 71, 2096, 73, 74
Offset: 1

Views

Author

Seiichi Manyama, Apr 25 2021

Keywords

Links

Crossrefs

Cf. A318636, A327124, A340626.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, Binomial[n/#, #] &, OddQ[#] &]; Array[a, 75] (* Amiram Eldar, Apr 25 2021 *)
  • PARI
    a(n) = sumdiv(n, d, (d%2)*binomial(n/d, d));
  • PARI
    N=99; x='x+O('x^N); Vec(sum(k=1, N, (1+x^k)^k-(1-x^k)^k)/2)

Formula

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

A376017 a(n) = Sum_{d|n} d^(n/d - d) * binomial(n/d,d).

Original entry on oeis.org

1, 2, 3, 5, 5, 12, 7, 32, 10, 90, 11, 264, 13, 686, 105, 1809, 17, 5166, 19, 11560, 2856, 28182, 23, 81456, 26, 159770, 61263, 375004, 29, 1122660, 31, 1984032, 1082598, 4456482, 560, 14486329, 37, 22413350, 16888053, 50674560, 41, 174582072, 43, 247627820, 241884450
Offset: 1

Views

Author

Seiichi Manyama, Sep 06 2024

Keywords

Links

Crossrefs

Cf. A318636.

Programs

  • PARI
    a(n) = sumdiv(n, d, d^(n/d-d)*binomial(n/d, d));
  • PARI
    my(N=50, x='x+O('x^N)); Vec(sum(k=1, N, x^k^2/(1-k*x^k)^(k+1)))
  • Python
    from math import comb
from itertools import takewhile
from sympy import divisors
def A376017(n): return sum(d**((m:=n//d)-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} x^(k^2) / (1 - k*x^k)^(k+1).
If p is prime, a(p) = p.
Showing 1-10 of 16 results. Next

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

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