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.

Previous Showing 11-16 of 16 results.

A308283 G.f. A(x) satisfies: A(x) = (1/(1 - x)) * A(x^3)*A(x^5)*A(x^7)* ... *A(x^(2*k-1))* ...

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 9, 11, 13, 17, 20, 25, 33, 39, 46, 59, 69, 83, 103, 121, 143, 175, 206, 242, 293, 342, 400, 481, 558, 650, 771, 893, 1040, 1225, 1414, 1634, 1912, 2202, 2538, 2954, 3390, 3893, 4517, 5165, 5913, 6820, 7783, 8893, 10212, 11626, 13239, 15156, 17219
Offset: 0

Views

Author

Ilya Gutkovskiy, May 18 2019

Keywords

Links

Crossrefs

Cf. A074206, A129374, A308271, A308284.

Programs

  • Mathematica
    terms = 55; A[] = 1; Do[A[x] = 1/(1 - x) Product[A[x^(2 k - 1)], {k, 2, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]

Formula

G.f.: Product_{k>=1} 1/(1 - x^(2*k-1))^A074206(2*k-1).

A321317 G.f. satisfies: A(x) = (1-x) * A(x^2)*A(x^3)*A(x^4)*...*A(x^n)*... .

Original entry on oeis.org

1, -1, -1, 0, -1, 2, -1, 3, -1, 1, 4, -2, -4, -1, 7, -11, 0, -6, 3, -6, 9, 2, -6, 2, 5, 12, -16, 48, 4, -1, -28, 26, -30, 30, -62, 12, 16, -55, -9, -39, 18, -47, 103, -149, 87, -182, 210, -9, 45, 75, 225, -174, 39, 273, 11, 164, -224, 77, -105, 117, -703, 715, -678
Offset: 0

Views

Author

Seiichi Manyama, Nov 04 2018

Keywords

Links

Crossrefs

Convolution inverse of A129374.
Cf. A074206, A129373, A321326.

Formula

G.f.: Product_{k>=1} (1 - x^k)^A074206(k) where A074206(n) is the number of ordered factorizations of n.

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).

A309174 E.g.f. A(x) satisfies: A(x) = (1/(1 - x)) * Product_{k>=2} A(x^k)^(1/k).

Original entry on oeis.org

1, 1, 3, 11, 65, 369, 3139, 24667, 268449, 2777345, 34932131, 432114891, 6790407073, 97969389361, 1671204338595, 28382893729499, 557174580764609, 10512263160373377, 228918738980395459, 4817409763554888715, 115117419384636141441, 2688602544800222293361
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 15 2019

Keywords

Crossrefs

Cf. A074206, A129374, A307604, A309175.

Programs

  • Mathematica
    terms = 21; A[] = 1; Do[A[x] = 1/(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/(1 - x^k)^(A074206(k)/k).

A385635 G.f. satisfies A(x) = x + Product_{n>=2} A(x^n) with A(0) = 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 4, 8, 8, 13, 15, 26, 26, 41, 48, 73, 80, 119, 136, 198, 225, 313, 367, 518, 585, 797, 941, 1264, 1466, 1953, 2285, 3022, 3524, 4571, 5391, 6993, 8152, 10440, 12316, 15684, 18370, 23236, 27327, 34389, 40364, 50370, 59292, 73880, 86547, 107080, 125976, 155266, 182058
Offset: 0

Views

Author

Paul D. Hanna, Jul 05 2025

Keywords

Examples

			G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 2*x^5 + 4*x^6 + 4*x^7 + 8*x^8 + 8*x^9 + 13*x^10 + 15*x^11 + 26*x^12 + ...
where
A(x) = x + A(x^2)*A(x^3)*A(x^4)*A(x^5)* ... * A(x^n) * ...

Links

Crossrefs

Cf. A129374, A129375.

Programs

  • PARI
    {a(n) = my(A=1+x +x*O(x^n)); for(i=1, ceil(log(n+2)/log(2)), A = x + prod(k=2,#A,subst(A, x, x^k)) +x*O(x^n); ); polcoef(A, n)}
for(n=0, 50, print1(a(n), ", "))

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 11-16 of 16 results.

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