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 1-3 of 3 results.

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

Crossrefs

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).

A363664 a(n) = Sum_{d|n} (n/d)^(n-n/d) * binomial(d+n-1,n).

Original entry on oeis.org

1, 4, 11, 56, 127, 1100, 1717, 19300, 64406, 383010, 352717, 23214660, 5200301, 191172406, 3465549077, 20859527460, 1166803111, 1010698826825, 17672631901, 102589250081802, 286539905316908, 75260204476154, 4116715363801, 548610025890719156
Offset: 1

Views

Author

Seiichi Manyama, Jun 14 2023

Keywords

Crossrefs

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^(n-n/#) * Binomial[# + n - 1, n] &]; Array[a, 25] (* Amiram Eldar, Jul 12 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^(n-n/d)*binomial(d+n-1, n));

Formula

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

A363666 a(n) = Sum_{d|n} (n/d)^(d-1) * binomial(d+n-2,n-1).

Original entry on oeis.org

1, 3, 7, 29, 71, 355, 925, 4425, 13276, 60111, 184757, 856357, 2704157, 12137147, 40367461, 176999505, 601080391, 2616894901, 9075135301, 38884056181, 138014377810, 583674491643, 2104098963721, 8823912454489, 32247616479976, 133998376789707
Offset: 1

Views

Author

Seiichi Manyama, Jun 14 2023

Keywords

Crossrefs

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^(#-1) * Binomial[# + n - 2, n - 1] &]; Array[a, 25] (* Amiram Eldar, Jul 12 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^(d-1)*binomial(d+n-2, n-1));

Formula

a(n) = [x^n] Sum_{k>0} x^k/(1 - k*x^k)^n.
Showing 1-3 of 3 results.