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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A302549 Expansion of Sum_{k>=1} (-1 + Product_{j>=1} 1/(1 - x^(k*j))^j).

Original entry on oeis.org

1, 4, 7, 17, 25, 58, 87, 177, 289, 528, 860, 1550, 2486, 4257, 6910, 11474, 18335, 29941, 47331, 75819, 118887, 187338, 290784, 452904, 696058, 1071234, 1632947, 2487504, 3759613, 5676424, 8512310, 12744903, 18975839, 28194293, 41691157, 61516394, 90379785
Offset: 1

Views

Author

Ilya Gutkovskiy, Jun 20 2018

Keywords

Comments

Inverse Moebius transform of A000219.

Links

Crossrefs

Cf. A000219, A047966, A047968, A300275, A302550.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n=0, 1, add(
      b(n-j)*numtheory[sigma][2](j), j=1..n)/n)
    end:
a:= n-> add(b(d), d=numtheory[divisors](n)):
seq(a(n), n=1..40);  # Alois P. Heinz, Jun 21 2018
  • Mathematica
    nmax = 37; Rest[CoefficientList[Series[Sum[-1 + Product[1/(1 - x^(k j))^j, {j, 1, nmax}], {k, 1, nmax}], {x, 0, nmax}], x]]
b[n_] := b[n] = SeriesCoefficient[Product[1/(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, 37}]
b[0] = 1; b[n_] := b[n] = Sum[b[n - j] DivisorSigma[2, j], {j, n}]/n; a[n_] := a[n] = Sum[b[d], {d, Divisors[n]}]; Table[a[n], {n, 37}]

Formula

G.f.: Sum_{k>=1} A000219(k)*x^k/(1 - x^k).
a(n) = Sum_{d|n} A000219(d).

A318290 Expansion of Sum_{k>=1} (-1 + Product_{j>=1} (1 + j*x^(k*j))).

Original entry on oeis.org

1, 3, 6, 10, 16, 33, 44, 74, 126, 204, 289, 503, 696, 1151, 1749, 2599, 3742, 5928, 8245, 12658, 18351, 26715, 37828, 55296, 78346, 111882, 159664, 226782, 315416, 446670, 618667, 860764, 1199995, 1649820, 2289020, 3157349, 4303996, 5878786, 8033272, 10894516, 14749052
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 23 2018

Keywords

Comments

Inverse Moebius transform of A022629.

Links

Crossrefs

Cf. A022629, A047966, A300278, A302550, A318025.

Programs

  • Maple
    a:=series(add(-1+mul(1+j*x^(k*j),j=1..100),k=1..100),x=0,42): seq(coeff(a,x,n),n=1..41); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 41; Rest[CoefficientList[Series[Sum[-1 + Product[(1 + j x^(k j)), {j, 1, nmax}], {k, 1, nmax}], {x, 0, nmax}], x]]
b[n_] := b[n] = SeriesCoefficient[Product[(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, 41}]
b[0] = 1; b[n_] := b[n] = Sum[Sum[(-d)^(k/d + 1), {d, Divisors[k]}] b[n - k], {k, 1, n}]/n; a[n_] := a[n] = Sum[b[d], {d, Divisors[n]}]; Table[a[n], {n, 41}]

Formula

G.f.: Sum_{k>=1} A022629(k)*x^k/(1 - x^k).
a(n) = Sum_{d|n} A022629(d).
Showing 1-2 of 2 results.

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