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.

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A302550 Expansion of Sum_{k>=1} (-1 + Product_{j>=1} (1 + x^(k*j))^j).

Original entry on oeis.org

1, 3, 6, 11, 17, 36, 50, 94, 148, 254, 386, 671, 1005, 1651, 2543, 4034, 6112, 9599, 14410, 22178, 33189, 50196, 74485, 111591, 164149, 242967, 355317, 520817, 755895, 1099219, 1584520, 2285960, 3275667, 4691845, 6682765, 9512213, 13471240, 19059192, 26851931, 37778822
Offset: 1

Views

Author

Ilya Gutkovskiy, Jun 20 2018

Keywords

Comments

Inverse Moebius transform of A026007.

Links

Crossrefs

Cf. A026007, A047966, A047968, A300276, A302549.

Programs

  • Maple
    with(numtheory):
b:= proc(n) option remember;
      add((-1)^(n/d+1)*d^2, d=divisors(n))
    end:
g:= proc(n) option remember;
      `if`(n=0, 1, add(b(k)*g(n-k), k=1..n)/n)
    end:
a:= n-> add(g(d), d=divisors(n)):
seq(a(n), n=1..40);  # Alois P. Heinz, Jun 21 2018
  • Mathematica
    nmax = 40; Rest[CoefficientList[Series[Sum[-1 + Product[(1 + x^(k j))^j, {j, 1, nmax}], {k, 1, nmax}], {x, 0, nmax}], x]]
b[n_] := b[n] = SeriesCoefficient[Product[(1 + x^k)^k , {k, 1, n}], {x, 0, n}]; a[n_] := a[n] = SeriesCoefficient[Sum[b[k] x^k/(1 - x^k), {k, 1, n}], {x, 0, n}]; Table[a[n], {n, 40}]
b[0] = 1; b[n_] := b[n] = Sum[Sum[(-1)^(j/d + 1) d^2, {d, Divisors[j]}] b[n - j], {j, n}]/n; a[n_] := a[n] = Sum[b[d], {d, Divisors[n]}]; Table[a[n], {n, 40}]

Formula

G.f.: Sum_{k>=1} A026007(k)*x^k/(1 - x^k).
a(n) = Sum_{d|n} A026007(d).

A318025 Expansion of Sum_{k>=1} (-1 + Product_{j>=1} 1/(1 - j*x^(k*j))).

Original entry on oeis.org

1, 4, 7, 18, 26, 66, 98, 216, 361, 701, 1171, 2287, 3763, 6887, 11707, 20740, 34637, 60678, 100581, 172609, 285924, 481671, 791317, 1323831, 2156856, 3561119, 5784021, 9459559, 15250217, 24783964, 39713789, 64032664, 102200203, 163617694, 259745174, 413886941, 653715969, 1035539948
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 23 2018

Keywords

Comments

Inverse Moebius transform of A006906.

Links

Crossrefs

Cf. A006906, A047968, A300277, A302549, A318290.

Programs

  • Mathematica
    nmax = 38; Rest[CoefficientList[Series[Sum[-1 + Product[1/(1 - j x^(k j)), {j, 1, nmax}], {k, 1, nmax}], {x, 0, nmax}], x]]
b[n_] := b[n] = SeriesCoefficient[Product[1/(1 - k x^k), {k, 1, n}], {x, 0, n}]; a[n_] := a[n] = SeriesCoefficient[Sum[b[k] x^k/(1 - x^k), {k, 1, n}], {x, 0, n}]; Table[a[n], {n, 38}]
Table[Sum[Total[Times @@@ IntegerPartitions[d]], {d, Divisors[n]}], {n, 38}]

Formula

G.f.: Sum_{k>=1} A006906(k)*x^k/(1 - x^k).
a(n) = Sum_{d|n} A006906(d).
Showing 1-2 of 2 results.

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