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.

A330047 Expansion of e.g.f. exp(-x) / (1 - sinh(x)).

Original entry on oeis.org

1, 0, 1, 3, 13, 75, 511, 4053, 36793, 375735, 4262971, 53203953, 724379173, 10684377795, 169713810631, 2888340723453, 52433443111153, 1011340189494255, 20654264750645491, 445249365444296553, 10103533212012216733, 240731286454287293115, 6008902898851584479551
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 28 2019

Keywords

Comments

Inverse binomial transform of A006154.

Links

Crossrefs

Cf. A006154, A009227, A052841, A330046, A346894, A346895, A346921.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[Exp[-x]/(1 - Sinh[x]), {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-(exp(x)-1)^2/2))) \\ Seiichi Manyama, May 07 2022
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (2*k)!*x^(2*k)/(2^k*prod(j=1, 2*k, 1-j*x)))) \\ Seiichi Manyama, May 07 2022
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, binomial(i, j)*stirling(j, 2, 2)*v[i-j+1])); v; \\ Seiichi Manyama, May 07 2022
  • PARI
    a(n) = sum(k=0, n\2, (2*k)!*stirling(n, 2*k, 2)/2^k); \\ Seiichi Manyama, May 07 2022

Formula

a(n) = Sum_{k=0..n} (-1)^(n - k) * binomial(n,k) * A006154(k).
a(n) ~ n! / ((2 + sqrt(2)) * (log(1 + sqrt(2)))^(n+1)). - Vaclav Kotesovec, Dec 03 2019
From Seiichi Manyama, May 07 2022: (Start)
E.g.f.: 1/(1 - (exp(x) - 1)^2 / 2).
G.f.: Sum_{k>=0} (2*k)! * x^(2*k)/(2^k * Product_{j=1..2*k} (1 - j * x)).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * Stirling2(k,2) * a(n-k).
a(n) = Sum_{k=0..floor(n/2)} (2*k)! * Stirling2(n,2*k)/2^k. (End)
a(0) = 1; a(n) = (-1)^n + Sum_{k=1..ceiling(n/2)} binomial(n,2*k-1) * a(n-2*k+1). - Prabha Sivaramannair, Oct 06 2023

A332258 E.g.f.: 1 / (1 + x - sinh(x)).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 20, 1, 112, 1681, 492, 27721, 371624, 319177, 13461604, 171387217, 319071456, 11466038689, 143550642140, 484491620089, 15758152572952, 199089883272217, 1077471975974484, 32827750137627457, 427744154995090256, 3385134777669637681
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 08 2020

Keywords

Comments

Number of labeled ordered partitions of an n-set into odd parts > 1.

Crossrefs

Cf. A006154, A009227, A032032.

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[1/(1 + x - Sinh[x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, 2 k - 1] a[n - 2 k + 1], {k, 2, Ceiling[n/2]}]; Table[a[n], {n, 0, 25}]
  • PARI
    seq(n)={Vec(serlaplace(1 / (1 + x - sinh(x + O(x*x^n)))))} \\ Andrew Howroyd, Feb 08 2020

Formula

a(0) = 1; a(n) = Sum_{k=2..ceiling(n/2)} binomial(n,2*k-1) * a(n-2*k+1).
a(n) ~ n! / ((cosh(r) - 1) * r^(n+1)), where r = 1.72911689821437486498840709347... is the root of the equation 1 + r - sinh(r) = 0. - Vaclav Kotesovec, Feb 08 2020

A009209 Expansion of e.g.f.: exp(sin(x))/exp(x).

Original entry on oeis.org

1, 0, 0, -1, 0, 1, 10, -1, -56, -279, 246, 4619, 14388, -53195, -556478, -864865, 13276912, 90192753, -72903378, -3987888493, -16957067028, 101506932205, 1411655530330, 2206092853799, -70455418153496, -549095655588183, 1428569363164230
Offset: 0

Views

Author

R. H. Hardin

Keywords

Crossrefs

Cf. A002017, A009227.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[Exp[Sin[x]]/Exp[x],{x,0,nn}], x]Range[0,nn]!] (* Harvey P. Dale, Jun 27 2012 *)
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(exp(sin(x))/exp(x))) \\ Michel Marcus, Apr 09 2022

Formula

a(0) = 1; a(n) = Sum_{k=1..floor((n-1)/2)} (-1)^k * binomial(n-1,2*k) * a(n-2*k-1). - Ilya Gutkovskiy, Apr 09 2022

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997
Definition clarified by Harvey P. Dale, Jun 27 2012

A009283 E.g.f.: exp(x + sinh(x)).

Original entry on oeis.org

1, 2, 4, 9, 24, 73, 246, 913, 3688, 16057, 74954, 372749, 1965156, 10942285, 64103006, 393902353, 2532010800, 16982676561, 118600412626, 860680689429, 6478753957948, 50505684285301, 407133297257542, 3389160344023385, 29098108436107592, 257364794368638009
Offset: 0

Views

Author

R. H. Hardin

Keywords

Crossrefs

Cf. A003724, A009227, A009282.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[Exp[x+Sinh[x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Dec 16 2022 *)
  • PARI
    x='x+O('x^66); Vec(serlaplace(exp(x + sinh(x)))) /* Joerg Arndt, Sep 01 2012 */

Formula

a(0) = 1; a(n) = a(n-1) + Sum_{k=0..floor((n-1)/2)} binomial(n-1,2*k) * a(n-2*k-1). - Ilya Gutkovskiy, Apr 10 2022

Extensions

Extended and signs tested by Olivier Gérard, Mar 15 1997
Name corrected by Arkadiusz Wesolowski, Sep 01 2012

A365893 Expansion of e.g.f. exp( Sum_{k>=0} x^(5*k+3) / (5*k+3)! ).

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 10, 0, 1, 280, 0, 165, 15400, 1, 30030, 1401400, 6995, 6806800, 190590401, 6506835, 1939938000, 36212380820, 4940624150, 687126039601, 9163671323015, 3761116975000, 297754623925175, 2982764271647875, 3067236941769001
Offset: 0

Views

Author

Seiichi Manyama, Sep 22 2023

Keywords

Links

Crossrefs

Cf. A006505, A009227, A307978.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=0, N\5, x^(5*k+3)/(5*k+3)!))))

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-3)/5)} binomial(n-1,5*k+2) * a(n-5*k-3).
Showing 1-5 of 5 results.

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