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.

Previous Showing 11-17 of 17 results.

A087138 Expansion of (1-sqrt(1-4*log(1+x)))/2.

Original entry on oeis.org

1, 1, 8, 64, 824, 12968, 252720, 5789712, 153169440, 4589004192, 153643615872, 5684390364288, 230307823878144, 10141452865049088, 482259966649655808, 24630247225278881280, 1344614199041549399040, 78137673004382654223360
Offset: 1

Views

Author

Vladeta Jovovic, Oct 18 2003

Keywords

Links

Crossrefs

Cf. A000108, A052851, A052803, A052886, A052895, A006531, A048287, A293604.
Cf. A370462, A370463.

Programs

  • Mathematica
    Rest[CoefficientList[Series[(1-Sqrt[1-4*Log[1+x]])/2, {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, May 03 2015 *)
  • PARI
    x='x+O('x^50); Vec(serlaplace((1-sqrt(1-4*log(1+x)))/2)) \\ G. C. Greubel, May 24 2017

Formula

a(n) = Sum_{k=1..n} Stirling1(n, k)*k!*Catalan(k-1).
a(n) ~ n! / (2*exp(1/8)*sqrt(Pi) * (exp(1/4)-1)^(n-1/2) * n^(3/2)). - Vaclav Kotesovec, May 03 2015
From Seiichi Manyama, Sep 09 2024: (Start)
E.g.f. satisfies A(x) = (log(1 + x)) / (1 - A(x)).
E.g.f.: Series_Reversion( exp(x * (1 - x)) - 1 ). (End)

A351906 Expansion of e.g.f. exp(x * (1 - x^4)).

Original entry on oeis.org

1, 1, 1, 1, 1, -119, -719, -2519, -6719, -15119, 1784161, 19902961, 119655361, 518763961, 1815974161, -212497445159, -3472602456959, -29605333299359, -177764320560959, -844590032480159, 97992221659873921, 2116963290135836521, 23379513665735470321
Offset: 0

Views

Author

Seiichi Manyama, Feb 25 2022

Keywords

Crossrefs

Cf. A293604, A246607, A351905.
Cf. A190877, A293507.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(x*(1-x^4))))
  • PARI
    a(n) = n!*sum(k=0, n\5, (-1)^k*binomial(n-4*k, k)/(n-4*k)!);
  • PARI
    a(n) = if(n<5, 1, a(n-1)-5!*binomial(n-1, 4)*a(n-5));

Formula

a(n) = n! * Sum_{k=0..floor(n/5)} (-1)^k * binomial(n-4*k,k)/(n-4*k)!.
a(n) = a(n-1) - 5! * binomial(n-1,4) * a(n-5) for n > 4.

A351905 Expansion of e.g.f. exp(x * (1 - x^3)).

Original entry on oeis.org

1, 1, 1, 1, -23, -119, -359, -839, 18481, 178417, 902161, 3318481, -69866279, -1011908039, -7204341143, -36194591159, 726745175521, 14326789219681, 131901636673441, 840736509931297, -16060449291985079, -408041402342457239, -4618341644958693959, -35691963052019431079
Offset: 0

Views

Author

Seiichi Manyama, Feb 25 2022

Keywords

Crossrefs

Cf. A293604, A246607, A351906.
Cf. A190875, A293493.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(x*(1-x^3))))
  • PARI
    a(n) = n!*sum(k=0, n\4, (-1)^k*binomial(n-3*k, k)/(n-3*k)!);
  • PARI
    a(n) = if(n<4, 1, a(n-1)-4!*binomial(n-1, 3)*a(n-4));

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} (-1)^k * binomial(n-3*k,k)/(n-3*k)!.
a(n) = a(n-1) - 4! * binomial(n-1,3) * a(n-4) for n > 3.

A377958 Expansion of e.g.f. exp(x - x^2)/(1 - x).

Original entry on oeis.org

1, 2, 3, 4, 17, 126, 787, 5048, 39489, 361882, 3641411, 39948492, 478777873, 6226077014, 87182747667, 1307703873856, 20922694556417, 355686434950578, 6402375749061379, 121645136562423572, 2432901971620591761, 51090940751194252462, 1124000727777806326163
Offset: 0

Views

Author

Seiichi Manyama, Nov 12 2024

Keywords

Crossrefs

Cf. A293604, A377959, A377960.
Cf. A018191.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, binomial(n-2*k, n-k)/k!);

Formula

a(n) = n! * Sum_{k=0..n} binomial(n-2*k,n-k) / k!.
a(n) = (n+1)*a(n-1) - 3*(n-1)*a(n-2) + 2*(n-1)*(n-2)*a(n-3) for n > 2.

A377959 Expansion of e.g.f. exp(x - x^2)/(1 - x)^2.

Original entry on oeis.org

1, 3, 9, 31, 141, 831, 5773, 45459, 403161, 3990331, 43544721, 518940423, 6706062949, 93404895351, 1394851282581, 22230473112571, 376610264357553, 6758060929028979, 128047472471583001, 2554547113522500591, 53523844242070603581, 1175091669834676927663
Offset: 0

Views

Author

Seiichi Manyama, Nov 12 2024

Keywords

Crossrefs

Cf. A293604, A377958, A377960.
Cf. A377954.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, binomial(n-2*k+1, n-k)/k!);

Formula

a(n) = n! * Sum_{k=0..n} binomial(n-2*k+1,n-k) / k!.
a(n) = (n+2)*a(n-1) - 3*(n-1)*a(n-2) + 2*(n-1)*(n-2)*a(n-3) for n > 2.

A377960 Expansion of e.g.f. exp(x - x^2)/(1 - x)^3.

Original entry on oeis.org

1, 4, 17, 82, 469, 3176, 24829, 219262, 2157257, 23405644, 277601161, 3572553194, 49576701277, 737902011952, 11725479449909, 198112664861206, 3546412902136849, 67047080265355412, 1334894917247980417, 27917550541234128514, 611874855066753173861, 14024463626236493578744
Offset: 0

Views

Author

Seiichi Manyama, Nov 12 2024

Keywords

Crossrefs

Cf. A293604, A377958, A377959.
Cf. A377955.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, binomial(n-2*k+2, n-k)/k!);

Formula

a(n) = n! * Sum_{k=0..n} binomial(n-2*k+2,n-k) / k!.
a(n) = (n+3)*a(n-1) - 3*(n-1)*a(n-2) + 2*(n-1)*(n-2)*a(n-3) for n > 2.

A374364 Expansion of e.g.f. exp( x - Sum_{k>=1} x^(2^k)/2^k ).

Original entry on oeis.org

1, 1, 0, -2, -8, -24, 16, 400, -3072, -38528, -18944, 1287936, 17843200, 149045248, -188786688, -12007184384, -1265929355264, -20275964313600, 3871935889408, 2355175169523712, 45658709327609856, 565591105847689216, -1448855443865600000
Offset: 0

Views

Author

Seiichi Manyama, Jul 06 2024

Keywords

Links

Crossrefs

Cf. A005388, A008683, A117209, A293604.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x-sum(k=1, ceil(log(N+1)/log(2)), x^2^k/2^k))))

Formula

E.g.f.: Product_{k>=1} (1 + x^(2*k-1))^(mu(2*k-1)/(2*k-1)), where mu() is the Moebius function.
Previous Showing 11-17 of 17 results.

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