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 1-5 of 5 results.

A292952 E.g.f.: exp(-x * exp(x)).

Original entry on oeis.org

1, -1, -1, 2, 9, 4, -95, -414, 49, 10088, 55521, 13870, -2024759, -15787188, -28612415, 616876274, 7476967905, 32522642896, -209513308607, -4924388011050, -38993940088199, -11731860520780, 3807154270837281, 52018152493218010, 278413297030360273
Offset: 0

Views

Author

Seiichi Manyama, Sep 27 2017

Keywords

Links

Crossrefs

Column k=1 of A293015.
Cf. this sequence (k=1), A292953 (k=2), A292954 (k=3), A292955 (k=4).
Cf. A003725.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[Exp[-x Exp[x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 15 2023 *)
  • PARI
    x='x+O('x^66); Vec(serlaplace(exp(-x*exp(x))))

Formula

a(n) = (-1)^n * A003725(n).

A292954 E.g.f.: exp(-1/3! * x^3 * exp(x)).

Original entry on oeis.org

1, 0, 0, -1, -4, -10, -10, 105, 1064, 6356, 25080, 9075, -1056660, -13219206, -106106364, -548948855, 139658960, 48411569800, 761039099824, 7815284148711, 52216924707660, 9385130453790, -6650556642220260, -132749143322588331, -1713641693856894824
Offset: 0

Views

Author

Seiichi Manyama, Sep 27 2017

Keywords

Links

Crossrefs

Column k=3 of A293015.
Cf. A292952 (k=1), A292953 (k=2), this sequence (k=3), A292955 (k=4).
Cf. A292910.

Programs

  • PARI
    x='x+O('x^66); Vec(serlaplace(exp(-1/3!*x^3*exp(x))))

Formula

a(n) = (-1)^n * A292910(n).

A292955 E.g.f.: exp(-1/4! * x^4 * exp(x)).

Original entry on oeis.org

1, 0, 0, 0, -1, -5, -15, -35, -35, 504, 6090, 45870, 270930, 1215500, 1995994, -42118440, -733409495, -8069463780, -70153266240, -468024155016, -1498366231020, 21132982355940, 568009017066260, 8607952077741940, 101448276642079059, 937291639168833850
Offset: 0

Views

Author

Seiichi Manyama, Sep 27 2017

Keywords

Links

Crossrefs

Column k=4 of A293015.
Cf. A292952 (k=1), A292953 (k=2), A292954 (k=3), this sequence (k=4).

Programs

  • PARI
    x='x+O('x^66); Vec(serlaplace(exp(-1/4!*x^4*exp(x))))

A293015 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where A(0,k) = 1 and A(n,k) = - Sum_{i=0..n-1} binomial(n-1,i) * binomial(i+1,k) * A(n-1-i,k) for n > 0.

Original entry on oeis.org

1, 1, -1, 1, -1, 0, 1, 0, -1, 1, 1, 0, -1, 2, 1, 1, 0, 0, -3, 9, -2, 1, 0, 0, -1, -3, 4, -9, 1, 0, 0, 0, -4, 20, -95, -9, 1, 0, 0, 0, -1, -10, 150, -414, 50, 1, 0, 0, 0, 0, -5, -10, 504, 49, 267, 1, 0, 0, 0, 0, -1, -15, 105, -343, 10088, 413, 1, 0, 0, 0, 0, 0, -6
Offset: 0

Views

Author

Seiichi Manyama, Sep 28 2017

Keywords

Examples

			Square array begins:
    1,  1,  1,  1,  1, ...
   -1, -1,  0,  0,  0, ...
    0, -1, -1,  0,  0, ...
    1,  2, -3, -1,  0, ...
    1,  9, -3, -4, -1, ...

Links

Crossrefs

Columns k=0-4 give: A000587, A292952, A292953, A292954, A292955.
Rows n=0 gives A000012.
Cf. A145460.

A346753 Expansion of e.g.f. -log( 1 - x^2 * exp(x) / 2 ).

Original entry on oeis.org

0, 0, 1, 3, 9, 40, 225, 1491, 11578, 102852, 1026945, 11394955, 139091106, 1852061718, 26716291693, 415033647315, 6908006807640, 122645325067576, 2313546734841633, 46209268921868595, 974228913850588750, 21620679147700290210, 503810188866302511501
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 01 2021

Keywords

Crossrefs

Cf. A000217, A009444, A133189, A292953, A305990, A346750, A346754, A346755.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[-Log[1 - x^2 Exp[x]/2], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 0; a[n_] := a[n] = Binomial[n, 2] + (1/n) Sum[Binomial[n, k] Binomial[n - k, 2] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 22}]

Formula

a(0) = 0; a(n) = binomial(n,2) + (1/n) * Sum_{k=1..n-1} binomial(n,k) * binomial(n-k,2) * k * a(k).
a(n) ~ (n-1)! / (2*LambertW(1/sqrt(2)))^n. - Vaclav Kotesovec, Aug 08 2021
a(n) = n! * Sum_{k=1..floor(n/2)} k^(n-2*k-1)/(2^k * (n-2*k)!). - Seiichi Manyama, Dec 14 2023
Showing 1-5 of 5 results.

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