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 1-3 of 3 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

A327034 Expansion of e.g.f. exp(x) / (2 - cosh(x)).

Original entry on oeis.org

1, 1, 2, 4, 14, 46, 242, 1114, 7814, 46246, 405482, 2933074, 30860414, 263817646, 3238391522, 31943268634, 448122565814, 5009616448246, 79063212894362, 987840438629794, 17322647732052014, 239217148602642046, 4614370558369770002, 69790939492563608554
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 28 2019

Keywords

Crossrefs

Cf. A081557, A094088, A330046.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[Exp[x]/(2 - Cosh[x]), {x, 0, nmax}], x] Range[0, nmax]!

Formula

a(0) = 1; a(n) = Sum_{k=1..n} (-1)^k * binomial(n,k) * (2^(k - 1) - 2) * a(n-k).
a(n) ~ n! * (7 + 4*sqrt(3) + (-1)^n) / ((3+2*sqrt(3)) * (log(2+sqrt(3)))^(n+1)). - Vaclav Kotesovec, Dec 03 2019

A348580 Expansion of e.g.f. exp(x) / (1 - sin(x)).

Original entry on oeis.org

1, 2, 5, 15, 53, 217, 1015, 5355, 31513, 204857, 1458875, 11299695, 94600373, 851419597, 8198959735, 84124450035, 916270051633, 10559066809937, 128362804540595, 1641730799916375, 22037407161945293, 309782122281453877, 4551072446448773455, 69747642031977698715
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 24 2021

Keywords

Links

Crossrefs

Cf. A000111, A000667, A186364, A330046, A348587.

Programs

  • Maple
    b:= proc(u, o) option remember; `if`(u+o=0, 1,
      add(b(o-1+j, u-j), j=1..u))
    end:
a:= n-> add(binomial(n, k)*b(k+1, 0), k=0..n):
seq(a(n), n=0..23);  # Alois P. Heinz, Oct 24 2021
  • Mathematica
    nmax = 23; CoefficientList[Series[Exp[x]/(1 - Sin[x]), {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    my(x='x+O('x^40)); Vec(serlaplace(exp(x)/(1-sin(x)))) \\ Michel Marcus, Oct 24 2021

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * A000111(k+1).
a(n) ~ 2^(n + 7/2) * n^(n + 3/2) / (Pi^(n + 3/2) * exp(n - Pi/2)). - Vaclav Kotesovec, Oct 25 2021
Showing 1-3 of 3 results.

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