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 1-4 of 4 results.

A318249 a(n) = (n - 1)! * d(n), where d(n) = number of divisors of n (A000005).

Original entry on oeis.org

1, 2, 4, 18, 48, 480, 1440, 20160, 120960, 1451520, 7257600, 239500800, 958003200, 24908083200, 348713164800, 6538371840000, 41845579776000, 2134124568576000, 12804747411456000, 729870602452992000, 9731608032706560000, 204363768686837760000, 2248001455555215360000, 206816133911079813120000
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 22 2018

Keywords

Links

Crossrefs

Column 1 of A338805.
Cf. A000005, A028342, A038048, A318250, A338814.

Programs

  • Mathematica
    Table[(n - 1)! DivisorSigma[0, n], {n, 1, 24}]
nmax = 24; Rest[CoefficientList[Series[Sum[Sum[x^(j k)/(j k), {j, 1, nmax}], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!]
nmax = 24; Rest[CoefficientList[Series[-Log[Product[(1 - x^k)^(1/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!]
  • PARI
    a(n) = (n-1)!*numdiv(n); \\ Michel Marcus, Aug 22 2018

Formula

E.g.f.: Sum_{k>=1} Sum_{j>=1} x^(j*k)/(j*k).
E.g.f.: -log(Product_{k>=1} (1 - x^k)^(1/k)).
E.g.f.: A(x) = log(B(x)), where B(x) = e.g.f. of A028342.
a(p^k) = (k + 1)*(p^k - 1)!, where p is a prime.

A330449 Expansion of e.g.f. Sum_{k>=1} (exp(x) - 1)^k / (k * (1 - (exp(x) - 1)^k)^2).

Original entry on oeis.org

1, 6, 36, 282, 2460, 25506, 299796, 3921882, 56977740, 913248786, 15917884356, 299358495882, 6066180049020, 131932872768066, 3057940695635316, 75151035318996282, 1954299203147952300, 53684552455571903346, 1553161560008013680676, 47162101103528811791082
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 15 2019

Keywords

Links

Crossrefs

Cf. A000219, A001157, A008277, A064602, A306046, A318250, A330353, A330450.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Sum[(Exp[x] - 1)^k/(k (1 - (Exp[x] - 1)^k)^2), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[Sum[StirlingS2[n, k] (k - 1)! DivisorSigma[2, k], {k, 1, n}], {n, 1, 20}]

Formula

E.g.f.: -Sum_{k>=1} k * log(1 - (exp(x) - 1)^k).
E.g.f.: A(x) = log(B(x)), where B(x) = e.g.f. of A306046.
G.f.: Sum_{k>=1} (k - 1)! * sigma_2(k) * x^k / Product_{j=1..k} (1 - j*x), where sigma_2 = A001157.
exp(Sum_{n>=1} a(n) * log(1 + x)^n / n!) = g.f. of A000219.
a(n) = Sum_{k=1..n} Stirling2(n,k) * (k - 1)! * sigma_2(k).
a(n) ~ n! * zeta(3) * n / (4 * (log(2))^(n+2)). - Vaclav Kotesovec, Dec 15 2019

A330450 Expansion of e.g.f. Sum_{k>=1} log(1 + x)^k / (k * (1 - log(1 + x)^k)^2).

Original entry on oeis.org

1, 4, 7, 55, -162, 4100, -49030, 779914, -11928008, 198650880, -3538477560, 70414760136, -1571134087824, 38788172175072, -1028732373217200, 28631225505910224, -826097667884640768, 24664145505337921920, -765245501125015575168, 24841409653689047496576
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 15 2019

Keywords

Links

Crossrefs

Cf. A000219, A001157, A008275, A318250, A330354, A330449, A330495.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Sum[Log[1 + x]^k/(k (1 - Log[1 + x]^k)^2), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[Sum[StirlingS1[n, k] (k - 1)! DivisorSigma[2, k], {k, 1, n}], {n, 1, 20}]

Formula

E.g.f.: -Sum_{k>=1} k * log(1 - log(1 + x)^k).
E.g.f.: log(Product_{k>=1} 1 / (1 - log(1 + x)^k)^k).
exp(Sum_{n>=1} a(n) * (exp(x) - 1)^n / n!) = g.f. of A000219.
a(n) = Sum_{k=1..n} Stirling1(n,k) * (k - 1)! * sigma_2(k), where sigma_2 = A001157.
Conjecture: a(n) ~ n! * (-1)^n * zeta(3) * n * exp(n) / (8 * (exp(1) - 1)^(n+2)). - Vaclav Kotesovec, Dec 16 2019

A318580 Expansion of e.g.f. exp(-1 + Product_{k>=1} 1/(1 - x^k)^k).

Original entry on oeis.org

1, 1, 7, 55, 601, 7561, 116191, 1999327, 39267985, 850964401, 20332107991, 527930427751, 14838001344937, 447653776595065, 14440021169407471, 495398956418435791, 18012260306904120481, 691502230924473978337, 27948692251661337581095, 1185878351946613955122711
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 29 2018

Keywords

Crossrefs

Cf. A000219, A001157, A058892, A318250.

Programs

  • Maple
    seq(n!*coeff(series(exp(-1+mul(1/(1-x^k)^k,k=1..100)),x=0,20),x,n),n=0..19); # Paolo P. Lava, Jan 09 2019
  • Mathematica
    nmax = 19; CoefficientList[Series[Exp[-1 + Product[1/(1 - x^k)^k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 19; CoefficientList[Series[Exp[-1 + Exp[Sum[DivisorSigma[2, k] x^k/k, {k, 1, nmax}]]], {x, 0, nmax}], x] Range[0, nmax]!
p[n_] := p[n] = Sum[DivisorSigma[2, k] p[n - k], {k, n}]/n; p[0] = 1; a[n_] := a[n] = Sum[p[k] k! Binomial[n - 1, k - 1] a[n - k], {k, n}]; a[0] = 1; Table[a[n], {n, 0, 19}]

Formula

E.g.f.: exp(-1 + exp(Sum_{k>=1} sigma_2(k)*x^k/k)).
E.g.f.: A(x) = exp(B(x) - 1), where B(x) = o.g.f. of A000219.
a(0) = 1; a(n) = Sum_{k=1..n} A000219(k)*k!*binomial(n-1,k-1)*a(n-k).
Showing 1-4 of 4 results.

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