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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A307596 G.f. A(x) satisfies: A(x) = (1/(1 + x)) * A(x^2)*A(x^3)*A(x^4)* ... *A(x^k)* ...

Original entry on oeis.org

1, -1, 0, -1, 0, 0, -1, 2, -1, 2, 0, 3, -4, 5, -1, 1, -3, 0, -1, -4, -1, -2, 3, -12, 6, -14, 15, -8, 17, -16, 25, -18, 23, 0, 5, 4, 15, 3, -12, 29, -29, 41, -59, 54, -56, 5, -89, 68, -110, 84, -137, 55, -52, 55, -95, 104, -53, -9, 47, -11, 109, -25, 157, -139, 278, -144, 455, -359, 413, -289, 554
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 17 2019

Keywords

Comments

Convolution inverse of A129373.

Examples

			G.f.: A(x) = 1 - x - x^3 - x^6 + 2*x^7 - x^8 + 2*x^9 + 3*x^11 - 4*x^12 + 5*x^13 - x^14 + ...

Links

Crossrefs

Cf. A074206, A129373, A129374, A318767, A321317.

Programs

  • Mathematica
    terms = 70; A[] = 1; Do[A[x] = 1/(1 + x) Product[A[x^k], {k, 2, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]

Formula

G.f.: Product_{k>=1} 1/(1 + x^k)^A074206(k).

A309175 E.g.f. A(x) satisfies: A(x) = (1 + x) * Product_{k>=2} A(x^k)^(1/k).

Original entry on oeis.org

1, 1, 1, 5, 17, 89, 769, 5389, 36385, 448433, 5058881, 51656021, 792623281, 10548735625, 141646268737, 2746494730589, 42803661694529, 688682904155489, 16532731961057665, 317686449365041573, 6296248862732350801, 159762846613265335481, 3488151227855502313601
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 15 2019

Keywords

Crossrefs

Cf. A074206, A129373, A307605, A309174.

Programs

  • Mathematica
    terms = 22; A[] = 1; Do[A[x] = (1 + x) Product[A[x^k]^(1/k), {k, 2, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x] Range[0, terms]!

Formula

E.g.f.: Product_{k>=1} (1 + x^k)^(A074206(k)/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).
Previous Showing 11-13 of 13 results.

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