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 1-3 of 3 results.

A346547 E.g.f.: Product_{k>=1} 1 / (1 - x^k)^exp(x).

Original entry on oeis.org

1, 1, 6, 36, 282, 2575, 28075, 340809, 4657996, 69874305, 1145441713, 20279904337, 386803154474, 7874727448757, 170678885319787, 3919163707551187, 95029714996046680, 2424604353738271201, 64940619086990938317, 1820746123923294245293, 53328181409328560026038
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 16 2021

Keywords

Comments

Exponential transform of A002745.

Crossrefs

Cf. A000203, A002745, A053529, A346545, A346546, A346548.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Product[1/(1 - x^k)^Exp[x], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 20; CoefficientList[Series[Exp[Exp[x] Sum[DivisorSigma[1, k] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
A002745[n_] := Sum[Binomial[n, k] DivisorSigma[1, k] (k - 1)!, {k, 1, n}]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] A002745[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

Formula

E.g.f.: exp( exp(x) * Sum_{k>=1} sigma(k) * x^k / k ).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A002745(k) * a(n-k).

A346545 E.g.f.: Product_{k>=1} 1 / (1 - x^k)^(exp(x)/k).

Original entry on oeis.org

1, 1, 5, 26, 175, 1384, 12933, 135050, 1582901, 20380208, 286577757, 4352682256, 71247772121, 1244923243966, 23166410620637, 456940648889070, 9521696033968393, 208851154175983608, 4812156417656806393, 116112764199821653284, 2928658457243240595901, 77042063713731887400418
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 16 2021

Keywords

Comments

Exponential transform of A002746.

Crossrefs

Cf. A000005, A002746, A028342, A346546, A346547, A346548.

Programs

  • Mathematica
    nmax = 21; CoefficientList[Series[Product[1/(1 - x^k)^(Exp[x]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Exp[Exp[x] Sum[DivisorSigma[0, k] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
A002746[n_] := Sum[Binomial[n, k] DivisorSigma[0, k] (k - 1)!, {k, 1, n}]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] A002746[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]

Formula

E.g.f.: exp( exp(x) * Sum_{k>=1} d(k) * x^k / k ).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A002746(k) * a(n-k).

A346548 E.g.f.: Product_{k>=1} 1 / (1 - x^k)^exp(-x).

Original entry on oeis.org

1, 1, 2, 6, 42, 175, 2015, 10843, 157388, 1240377, 20118077, 172029231, 4052166250, 36360150385, 952965601471, 11194257455977, 316421367496344, 3722989943371217, 134504815853036649, 1641201826969536379, 67298415781492985366, 935342610632498431241, 40176825083871581430723
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 16 2021

Keywords

Comments

Exponential transform of A002743.
The first negative term is a(71) = -1.2234788... * 10^104.

Crossrefs

Cf. A000203, A002743, A053529, A346545, A346546, A346547.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[Product[1/(1 - x^k)^Exp[-x], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Exp[-x] Sum[DivisorSigma[1, k] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
A002743[n_] := Sum[(-1)^(n - k) Binomial[n, k] DivisorSigma[1, k] (k - 1)!, {k, 1, n}]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] A002743[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]

Formula

E.g.f.: exp( exp(-x) * Sum_{k>=1} sigma(k) * x^k / k ).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A002743(k) * a(n-k).
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.