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 1-7 of 7 results.

A277458 Expansion of e.g.f. -1/(1-LambertW(-x)).

Original entry on oeis.org

-1, 1, 0, 3, 16, 165, 2016, 30415, 539904, 11049129, 256038400, 6627314331, 189517916160, 5933803272397, 201893195083776, 7417376809230375, 292648536838045696, 12341039738944113105, 553942486234823786496, 26369048375194607316019, 1326864458454400696320000
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 16 2016

Keywords

Links

Crossrefs

Cf. A000312, A086331, A133297, A134095, A277510, A308506.

Programs

  • Maple
    seq(n!*add((-1)^(k+1)*k*n^(n-k-1)/(n-k)!, k = 1..n), n = 1..20); # Peter Bala, Jul 23 2021
  • Mathematica
    CoefficientList[Series[-1/(1-LambertW[-x]), {x, 0, 25}], x] * Range[0, 25]!
  • PARI
    my(x='x+O('x^50)); Vec(serlaplace(-1/(1 - lambertw(-x)))) \\ G. C. Greubel, Nov 07 2017

Formula

a(n) ~ n^(n-1) / 4.
a(n) = n!*Sum_{k = 1..n} (-1)^(k+1)*k*n^(n-k-1)/(n-k)! for n >= 1. Cf. A133297. - Peter Bala, Jul 23 2021
a(n) = (-1)^(n+1)*U(1-n, -n, -n) where U is the Kummer U function. - Peter Luschny, Jan 23 2025

A277489 Expansion of e.g.f. -LambertW(-log(1+x)).

Original entry on oeis.org

0, 1, 1, 5, 26, 224, 2244, 28496, 417976, 7122384, 136770960, 2937770472, 69626588976, 1806936836184, 50936933449752, 1550292926398680, 50661309325357824, 1769286989373994752, 65762170385201959680, 2591979585702305271552, 107982615297265761991680
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 17 2016

Keywords

Links

Crossrefs

Cf. A001865, A052807, A133297.
Cf. A357337, A357338.

Programs

  • Mathematica
    CoefficientList[Series[-LambertW[-Log[1+x]], {x, 0, 20}], x] * Range[0, 20]!
Table[Sum[StirlingS1[n, k]*k^(k-1), {k, 1, n}], {n, 0, 20}]
  • PARI
    my(x='x+O('x^50)); concat([0], Vec(serlaplace(-lambertw(-log(1+x))))) \\ G. C. Greubel, Jun 21 2017

Formula

a(n) = Sum_{k=1..n} Stirling1(n, k)*k^(k-1).
a(n) ~ (exp(exp(-1))-1)^(1/2-n) * exp(-exp(-1)/2+1/2-n) * n^(n-1).
E.g.f.: Series_Reversion( exp(x * exp(-x)) - 1 ). - Seiichi Manyama, Sep 10 2024

A323673 Expansion of e.g.f. log(1 - LambertW(-x)*(2 + LambertW(-x))/2).

Original entry on oeis.org

0, 1, 0, 2, 7, 69, 696, 9400, 148506, 2753793, 58255840, 1388008566, 36768832200, 1072407094693, 34151921130432, 1179292944433500, 43892264744070736, 1751768399754149025, 74633720517351765504, 3380997879130123703818, 162286529338732345488000, 8227876237310253918100581
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 23 2019

Keywords

Crossrefs

Cf. A000272, A001858, A001865, A133297.

Programs

  • Maple
    seq(n!*coeff(series(log(1-LambertW(-x)*(2+LambertW(-x))/2),x=0,22),x,n),n=0..21); # Paolo P. Lava, Jan 28 2019
  • Mathematica
    nmax = 21; CoefficientList[Series[Log[1 - LambertW[-x] (2 + LambertW[-x])/2], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = n^(n - 2) - Sum[Binomial[n, k] (n - k)^(n - k - 2) k a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 0, 21}]

Formula

E.g.f.: log(1 + Sum_{k>=1} k^(k-2)*x^k/k!).
a(0) = 0; a(n) = A000272(n) - (1/n)*Sum_{k=1..n-1} binomial(n,k)*A000272(n-k)*k*a(k).
a(n) ~ 2 * n^(n-2) / 3. - Vaclav Kotesovec, Jan 24 2019

A332236 E.g.f.: -log(2 - 1 / (1 + LambertW(-x))).

Original entry on oeis.org

1, 5, 41, 466, 6769, 119736, 2497585, 60037328, 1634619969, 49733223040, 1672657257721, 61636181886720, 2470033974057649, 106970912288285696, 4979259164362745025, 247940951411958163456, 13152705012933836446465, 740578125097986605678592, 44115815578591964641401289
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 07 2020

Keywords

Crossrefs

Cf. A000312, A001865, A133297, A202477, A210661, A308863, A323673, A332237.

Programs

  • Mathematica
    nmax = 19; CoefficientList[Series[-Log[2 - 1/(1 + LambertW[-x])], {x, 0, nmax}], x] Range[0, nmax]! // Rest
a[n_] := a[n] = n^n + (1/n) Sum[Binomial[n, k] (n - k)^(n - k) k a[k], {k, 1, n - 1}]; Table[a[n], {n, 1, 19}]

Formula

E.g.f.: -log(1 - Sum_{k>=1} k^k * x^k / k!).
a(n) = n^n + (1/n) * Sum_{k=1..n-1} binomial(n,k) * (n-k)^(n-k) * k * a(k).
a(n) ~ (n-1)! * 2^n * exp(n/2). - Vaclav Kotesovec, Feb 16 2020

A332237 E.g.f.: -log(1 + LambertW(-x) * (2 + LambertW(-x)) / 2).

Original entry on oeis.org

1, 2, 8, 49, 409, 4356, 56734, 877094, 15742521, 322454800, 7434673036, 190792267128, 5398552673617, 167087263076384, 5617979017621650, 203987454978218416, 7957053981454827601, 331920300203780633856, 14746208516909980554736, 695208730205550274544000
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 07 2020

Keywords

Crossrefs

Cf. A000272, A001858, A001865, A057817, A133297, A218688, A323673, A332236.

Programs

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

Formula

E.g.f.: -log(1 - Sum_{k>=1} k^(k-2) * x^k / k!).
a(n) = n^(n-2) + (1/n) * Sum_{k=1..n-1} binomial(n,k) * (n-k)^(n-k-2) * k * a(k).
a(n) ~ 2 * n^(n-2). - Vaclav Kotesovec, Feb 16 2020

A347993 a(n) = n! * Sum_{k=1..n} (-1)^(k+1) * n^(n-k) / (n-k)!.

Original entry on oeis.org

1, 2, 15, 136, 1645, 24336, 426979, 8658560, 199234809, 5128019200, 145969492471, 4552809182208, 154404454932325, 5656950010320896, 222655633595044875, 9369696305273798656, 419790650812640438641, 19950175280765680680960, 1002394352017754098219999, 53092232229227200348160000
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 23 2021

Keywords

Crossrefs

Cf. A000169, A000312, A001865, A063169, A133297, A134095, A277458, A347994.

Programs

  • Mathematica
    Table[n! Sum[(-1)^(k + 1) n^(n - k)/(n - k)!, {k, 1, n}], {n, 1, 20}]
nmax = 20; CoefficientList[Series[-LambertW[-x]/(1 - LambertW[-x]^2), {x, 0, nmax}], x] Range[0, nmax]! // Rest
  • PARI
    a(n) = n! * sum(k=1, n, (-1)^(k+1)*n^(n-k)/(n-k)!); \\ Michel Marcus, Sep 23 2021

Formula

E.g.f.: -LambertW(-x) / (1 - LambertW(-x)^2).
a(n) = n * A133297(n).

A347994 a(n) = n! * Sum_{k=1..n-1} (-1)^(k+1) * n^(n-k-2) / (n-k-1)!.

Original entry on oeis.org

0, 1, 4, 30, 296, 3720, 56652, 1014832, 20909520, 487198080, 12667470740, 363607605504, 11420819358456, 389646915374080, 14349217119054300, 567315485527234560, 23967624180805666208, 1077568488585047605248, 51369752823292604784420, 2588268388538639982592000
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 23 2021

Keywords

Crossrefs

Cf. A000169, A000435, A133297, A134095, A347993.

Programs

  • Mathematica
    Table[n! Sum[(-1)^(k + 1) n^(n - k - 2)/(n - k - 1)!, {k, 1, n - 1}], {n, 1, 20}]
nmax = 20; CoefficientList[Series[-LambertW[-x] - Log[1 - LambertW[-x]], {x, 0, nmax}], x] Range[0, nmax]! // Rest
  • PARI
    a(n) = n! * sum(k=1, n-1, (-1)^(k+1)*n^(n-k-2)/(n-k-1)!); \\ Michel Marcus, Sep 23 2021

Formula

E.g.f.: -LambertW(-x) - log(1 - LambertW(-x)).
a(n) = A134095(n) / n.
Showing 1-7 of 7 results.

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