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).

A292953 E.g.f.: exp(-1/2! * x^2 * exp(x)).

Original entry on oeis.org

1, 0, -1, -3, -3, 20, 150, 504, -343, -18180, -140220, -500500, 2032899, 50210082, 441768236, 1740141480, -13025325615, -330558552376, -3452606080848, -16648495695792, 136964192085395, 4315989335784630, 55121200672923924, 352945156766431592
Offset: 0

Views

Author

Seiichi Manyama, Sep 27 2017

Keywords

Links

Crossrefs

Column k=2 of A293015.
Cf. A292952 (k=1), this sequence (k=2), A292954 (k=3), A292955 (k=4).

Programs

  • PARI
    x='x+O('x^66); Vec(serlaplace(exp(-1/2!*x^2*exp(x))))

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).

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.

A346755 Expansion of e.g.f. -log( 1 - x^4 * exp(x) / 4! ).

Original entry on oeis.org

0, 0, 0, 0, 1, 5, 15, 35, 105, 756, 6510, 46530, 289245, 1892605, 16187171, 170721915, 1833783770, 18875258780, 196470797580, 2255939795436, 29179692064545, 401813199660285, 5612352516200815, 79620308330422475, 1182881543312932386
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 01 2021

Keywords

Crossrefs

Cf. A000332, A009444, A145454, A292955, A305990, A346752, A346753, A346754.

Programs

  • Mathematica
    nmax = 24; CoefficientList[Series[-Log[1 - x^4 Exp[x]/4!], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 0; a[n_] := a[n] = Binomial[n, 4] + (1/n) Sum[Binomial[n, k] Binomial[n - k, 4] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 24}]

Formula

a(0) = 0; a(n) = binomial(n,4) + (1/n) * Sum_{k=1..n-1} binomial(n,k) * binomial(n-k,4) * k * a(k).
a(n) ~ (n-1)! / (4*LambertW(3^(1/4)/2^(5/4)))^n. - Vaclav Kotesovec, Aug 08 2021
a(n) = n! * Sum_{k=1..floor(n/4)} k^(n-4*k-1)/(24^k * (n-4*k)!). - Seiichi Manyama, Dec 14 2023
Showing 1-5 of 5 results.

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

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