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 1-2 of 2 results.

A304963 Expansion of 1/(1 - Sum_{i>=1, j>=1, k>=1} x^(i*j*k)).

Original entry on oeis.org

1, 1, 4, 10, 31, 82, 241, 664, 1898, 5316, 15058, 42374, 119718, 337432, 952373, 2685906, 7578248, 21376331, 60306495, 170120330, 479922212, 1353855927, 3819280961, 10774233218, 30394408336, 85743168417, 241883489742, 682358211402, 1924947591447, 5430317571250, 15319043353639
Offset: 0

Views

Author

Ilya Gutkovskiy, May 22 2018

Keywords

Comments

Invert transform of A007425.

Links

Crossrefs

Cf. A000005, A007425, A011782, A129921, A174465, A280473, A304964.

Programs

  • Maple
    A:= proc(n, k) option remember; `if`(k=1, 1,
      add(A(d, k-1), d=numtheory[divisors](n)))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(A(j, 3)*a(n-j), j=1..n))
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, May 22 2018
  • Mathematica
    nmax = 30; CoefficientList[Series[1/(1 - Sum[x^(i j k), {i, 1, nmax}, {j, 1, nmax/i}, {k, 1, nmax/i/j}]), {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[1/(1 - Sum[Sum[DivisorSigma[0, d], {d, Divisors[k]}] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Sum[DivisorSigma[0, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 30}]

Formula

G.f.: 1/(1 - Sum_{k>=1} A007425(k)*x^k).

A305049 Expansion of 1/(1 - Sum_{k>=1} tau_k(k)*x^k), where tau_k(k) = number of ordered k-factorizations of k (A163767).

Original entry on oeis.org

1, 1, 3, 8, 27, 67, 216, 569, 1747, 4812, 14041, 39483, 115408, 326385, 941735, 2684170, 7725097, 22063737, 63354066, 181223899, 519883185, 1488316952, 4266788191, 12219763777, 35023995792, 100326757107, 287503501905, 823654031283, 2360146144917, 6761847714698, 19374935267810
Offset: 0

Views

Author

Ilya Gutkovskiy, May 24 2018

Keywords

Comments

Invert transform of A163767.

Links

Crossrefs

Cf. A001906, A055887, A088305, A129373, A129374, A129921, A163767, A304963, A304964, A304965.

Programs

  • Maple
    A:= proc(n, k) option remember; `if`(k=1, 1,
      add(A(d, k-1), d=numtheory[divisors](n)))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(A(j$2)*a(n-j), j=1..n))
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, May 24 2018
  • Mathematica
    nmax = 30; CoefficientList[Series[1/(1 - Sum[Times @@ (Binomial[# + k - 1, k - 1] & /@ FactorInteger[k][[All, 2]]) x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Times @@ (Binomial[# + k - 1, k - 1] & /@ FactorInteger[k][[All, 2]]) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 30}]

Formula

G.f.: 1/(1 - Sum_{k>=1} A163767(k)*x^k).
Showing 1-2 of 2 results.

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

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