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.

A385308 Expansion of e.g.f. 1/(1 - 2 * x * cosh(x))^(1/2).

Original entry on oeis.org

1, 1, 3, 18, 141, 1400, 17055, 245392, 4070073, 76483584, 1606033755, 37267953536, 947051118981, 26156846230528, 780174007426359, 24992424003517440, 855795857724702705, 31193844533488074752, 1205893835653392258867, 49280187764171870470144, 2122704756621224015194365
Offset: 0

Views

Author

Seiichi Manyama, Jun 24 2025

Keywords

Crossrefs

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
    a001147(n) = prod(k=0, n-1, 2*k+1);
    a(n) = sum(k=0, n, a001147(k)*a185951(n, k));

Formula

a(n) = Sum_{k=0..n} A001147(k) * A185951(n,k), where A185951(n,0) = 0^n.
a(n) ~ sqrt(2) * n^n / (sqrt(1 + r*sqrt(1 - 4*r^2)) * exp(n) * r^n), where r = 0.452787214835453627588998503316635625709288535855... is the root of the equation 2*r*cosh(r) = 1. - Vaclav Kotesovec, Jun 28 2025

A352646 Expansion of e.g.f. 1/(1 - 2 * x * cos(x)).

Original entry on oeis.org

1, 2, 8, 42, 288, 2410, 24000, 277186, 3648512, 53936082, 885150720, 15970846298, 314273439744, 6698574264122, 153746319720448, 3780677636321010, 99163499845386240, 2763481838977368994, 81542013760903053312, 2539717324111483027594
Offset: 0

Views

Author

Seiichi Manyama, Mar 25 2022

Keywords

Crossrefs

Programs

  • Mathematica
    With[{m = 19}, Range[0, m]! * CoefficientList[Series[1/(1 - 2*x*Cos[x]), {x, 0, m}], x]] (* Amiram Eldar, Mar 26 2022 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-2*x*cos(x))))
    
  • PARI
    a(n) = if(n==0, 1, 2*sum(k=0, (n-1)\2, (-1)^k*(2*k+1)*binomial(n, 2*k+1)*a(n-2*k-1)));

Formula

a(0) = 1; a(n) = 2 * Sum_{k=0..floor((n-1)/2)} (-1)^k * (2*k+1) * binomial(n,2*k+1) * a(n-2*k-1).
a(n) ~ n! / ((1 - r * sqrt(4*r^2 - 1)) * r^n), where r = A196603 = 0.6100312844641759753709630735134103246737209791121692378637516075328... is the root of the equation 2*r*cos(r) = 1. - Vaclav Kotesovec, Mar 27 2022
a(n) = Sum_{k=0..n} 2^k * k! * i^(n-k) * A185951(n,k), where i is the imaginary unit. - Seiichi Manyama, Jun 25 2025

A352649 Expansion of e.g.f. 1/(1 - 3 * x * cosh(x)).

Original entry on oeis.org

1, 3, 18, 171, 2160, 34035, 643680, 14203371, 358178688, 10161542691, 320315005440, 11106766229163, 420132741912576, 17216605635562515, 759789379494512640, 35925442734363182955, 1811923104577065615360, 97097117111612660889411, 5509300889675218610552832
Offset: 0

Views

Author

Seiichi Manyama, Mar 25 2022

Keywords

Crossrefs

Programs

  • Mathematica
    With[{m = 17}, Range[0, m]! * CoefficientList[Series[1/(1 - 3*x*Cosh[x]), {x, 0, m}], x]] (* Amiram Eldar, Mar 26 2022 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-3*x*cosh(x))))
    
  • PARI
    a(n) = if(n==0, 1, 3*sum(k=0, (n-1)\2, (2*k+1)*binomial(n, 2*k+1)*a(n-2*k-1)));

Formula

a(0) = 1; a(n) = 3 * Sum_{k=0..floor((n-1)/2)} (2*k+1) * binomial(n,2*k+1) * a(n-2*k-1).
a(n) = Sum_{k=0..n} 3^k * k! * A185951(n,k). - Seiichi Manyama, Jun 25 2025
Showing 1-3 of 3 results.