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 1-4 of 4 results.

A318636 Expansion of Sum_{n>=1} ( (1 + x^n)^n - 1 ).

Original entry on oeis.org

1, 2, 3, 5, 5, 9, 7, 14, 10, 20, 11, 31, 13, 35, 25, 45, 17, 74, 19, 70, 56, 77, 23, 161, 26, 104, 111, 154, 29, 261, 31, 222, 198, 170, 56, 536, 37, 209, 325, 496, 41, 623, 43, 605, 626, 299, 47, 1407, 50, 602, 731, 1092, 53, 1305, 517, 1443, 1026, 464, 59, 4002, 61, 527, 1429, 2381, 1352, 2596, 67, 3009, 1840, 2787, 71, 6719, 73, 740, 5378, 4655, 407, 5135, 79, 10118
Offset: 1

Views

Author

Paul D. Hanna, Sep 07 2018

Keywords

Examples

			G.f.: A(x) = x + 2*x^2 + 3*x^3 + 5*x^4 + 5*x^5 + 9*x^6 + 7*x^7 + 14*x^8 + 10*x^9 + 20*x^10 + 11*x^11 + 31*x^12 + 13*x^13 + 35*x^14 + 25*x^15 + 45*x^16 + ...
such that
A(x) = x + (1 + x^2)^2 - 1 + (1 + x^3)^3 - 1 + (1 + x^4)^4 - 1 + (1 + x^5)^5 - 1 + (1 + x^6)^6 - 1 + (1 + x^7)^7-1 + ...
RELATED SERIES.
The g.f. A(x) equals following series at y = 1:
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. A143862, A318637, A318638.

Programs

  • PARI
    {a(n) = polcoeff( sum(m=1,n, (x^m + 1 +x*O(x^n))^m - 1), n)}
for(n=1,100, print1(a(n),", "))

Formula

a(n) = Sum_{d|n} binomial(n/d,d). - Ridouane Oudra, May 02 2019
G.f.: Sum_{k >=1} x^(k^2)/(1-x^k)^(k+1). - Seiichi Manyama, Oct 30 2023

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

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).
Showing 1-4 of 4 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.