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 1-8 of 8 results.

A363647 Expansion of Sum_{k>0} (1/(1 - (k*x)^k)^3 - 1).

Original entry on oeis.org

3, 18, 91, 879, 9396, 145010, 2470665, 50728749, 1162458352, 30058615320, 855935011911, 26761537897338, 908625319776864, 33340089815701086, 1313681976619686558, 55341921135416377497, 2481720785659010292702, 118040125809311823596960
Offset: 1

Views

Author

Seiichi Manyama, Jun 13 2023

Keywords

Crossrefs

Cf. A023887, A338663, A363646, A363648.
Cf. A363639.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^n * Binomial[# + 2, 2] &]; Array[a, 20] (* Amiram Eldar, Jul 17 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^n*binomial(d+2, 2));

Formula

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

A363642 Expansion of Sum_{k>0} x^k/(1 - k*x^k)^3.

Original entry on oeis.org

1, 4, 7, 17, 16, 55, 29, 129, 100, 311, 67, 1135, 92, 1919, 1486, 5409, 154, 17038, 191, 33491, 20938, 67871, 277, 262861, 9701, 373127, 296110, 978727, 436, 3134821, 497, 5051969, 3898522, 10027655, 474146, 39352069, 704, 49808159, 48362926, 127403221, 862, 411286429, 947
Offset: 1

Views

Author

Seiichi Manyama, Jun 13 2023

Keywords

Crossrefs

Cf. A363639, A363643, A363644.
Cf. A167531, A363645.
Cf. A007437, A363650.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^(#-1) * Binomial[# + 1, 2] &]; Array[a, 50] (* Amiram Eldar, Jul 18 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^(d-1)*binomial(d+1, 2));

Formula

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

A362683 Expansion of Sum_{k>0} (1/(1 - k*x^k)^2 - 1).

Original entry on oeis.org

2, 7, 10, 25, 16, 78, 22, 153, 136, 298, 34, 1254, 40, 1214, 2004, 3825, 52, 11385, 58, 20894, 18932, 25006, 70, 150002, 18826, 115274, 199828, 389510, 88, 1334624, 94, 1725281, 2131188, 2360266, 725948, 14878299, 112, 10486958, 22329428, 37317986, 124, 120957336, 130
Offset: 1

Views

Author

Seiichi Manyama, Jun 13 2023

Keywords

Crossrefs

Cf. A338662, A363639, A363640.
Cf. A007503, A055225, A359103, A363646.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^# * (# + 1) &]; Array[a, 50] (* Amiram Eldar, Jul 17 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^d*(d+1));

Formula

a(n) = Sum_{d|n} (n/d)^d * (d+1) = A055225(n) + A359103(n).
If p is prime, a(p) = 1 + 3*p.

A363640 Expansion of Sum_{k>0} (1/(1 - k*x^k)^4 - 1).

Original entry on oeis.org

4, 18, 32, 91, 76, 358, 148, 917, 796, 2368, 408, 10354, 612, 16586, 16984, 52873, 1208, 180408, 1616, 374934, 271408, 749070, 2692, 3350370, 178376, 4592968, 4349008, 13197802, 5076, 45402484, 6108, 74470417, 64515400, 149432876, 10324768, 652324677, 10028
Offset: 1

Views

Author

Seiichi Manyama, Jun 13 2023

Keywords

Crossrefs

Cf. A338662, A362683, A363639.
Cf. A116963, A363648.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^# * Binomial[# + 3, 3] &]; Array[a, 40] (* Amiram Eldar, Jul 17 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^d*binomial(d+3, 3));

Formula

a(n) = Sum_{d|n} (n/d)^d * binomial(d+3,3).

A363643 Expansion of Sum_{k>0} x^(2*k)/(1 - k*x^k)^3.

Original entry on oeis.org

0, 1, 3, 7, 10, 22, 21, 53, 45, 126, 55, 373, 78, 764, 390, 2009, 136, 5995, 171, 12501, 5334, 28392, 253, 92353, 1550, 160070, 79110, 394913, 406, 1167406, 465, 2081649, 1083126, 4457010, 69650, 14534659, 666, 22414016, 13818246, 53200481, 820, 158661826, 903
Offset: 1

Views

Author

Seiichi Manyama, Jun 13 2023

Keywords

Crossrefs

Cf. A363639, A363642, A363644.
Cf. A363651.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^(#-2) * Binomial[#, 2] &]; Array[a, 50] (* Amiram Eldar, Jul 18 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^(d-2)*binomial(d, 2));

Formula

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

A363644 Expansion of Sum_{k>0} x^(3*k)/(1 - k*x^k)^3.

Original entry on oeis.org

0, 0, 1, 3, 6, 11, 15, 27, 29, 60, 45, 145, 66, 318, 146, 789, 120, 2199, 153, 4890, 1406, 11730, 231, 34175, 426, 67884, 20738, 163956, 378, 453341, 435, 882153, 295742, 1966608, 10230, 5656484, 630, 10027674, 3897938, 23069916, 780, 63648517, 861, 113050536
Offset: 1

Views

Author

Seiichi Manyama, Jun 13 2023

Keywords

Crossrefs

Cf. A363639, A363642, A363643.
Cf. A363610, A363652.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^(# - 3)*Binomial[# - 1, 2] &]; Array[a, 50] (* Amiram Eldar, Jul 18 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^(d-3)*binomial(d-1, 2));

Formula

a(n) = Sum_{d|n} (n/d)^(d-3) * binomial(d-1,2).

A363661 a(n) = Sum_{d|n} (n/d)^d * binomial(d+n,n).

Original entry on oeis.org

2, 12, 32, 150, 282, 1890, 3488, 21582, 54650, 282612, 705564, 4072224, 10400782, 55006530, 158987232, 790611350, 2333606526, 11573213196, 35345264180, 168673694070, 540848064614, 2500462200182, 8233430728152, 37445946291600, 126411051769652
Offset: 1

Views

Author

Seiichi Manyama, Jun 14 2023

Keywords

Comments

All terms are even. - Robert Israel, Nov 23 2023

Links

Crossrefs

Cf. A362683, A363639, A363640.
Cf. A055225, A363660, A363662, A363663.

Programs

  • Maple
    f:= proc(n) local d;
   add((n/d)^d * binomial(n+d,n), d = numtheory:-divisors(n))
end proc:
map(f, [$1..30]); # Robert Israel, Nov 23 2023
  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^# * Binomial[# + n, n] &]; Array[a, 30] (* Amiram Eldar, Jul 05 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^d*binomial(d+n, n));

Formula

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

A363668 a(n) = Sum_{d|n} (n/d)^d * binomial(d+n-1,d).

Original entry on oeis.org

1, 7, 19, 91, 151, 1135, 1765, 12355, 28846, 157917, 352837, 2280955, 5200469, 29986201, 80469589, 427061795, 1166803399, 6211188028, 17672632261, 89483074521, 271071666724, 1316291647997, 4116715364329, 19595444140771, 63205674328876, 292318539358879
Offset: 1

Views

Author

Seiichi Manyama, Jun 14 2023

Keywords

Crossrefs

Cf. A055225, A362683, A363639, A363640.
Cf. A343549, A363669.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^# * Binomial[# + n - 1, #] &]; Array[a, 30] (* Amiram Eldar, Jul 12 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^d*binomial(d+n-1, d));

Formula

a(n) = [x^n] Sum_{k>0} (1/(1 - k*x^k)^n - 1).
Showing 1-8 of 8 results.

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