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.

A368767 a(n) = n! * (1 + Sum_{k=0..n} (-1)^k * binomial(k+2,3) / k!).

Original entry on oeis.org

1, 0, 4, 2, 28, 105, 686, 4718, 37864, 340611, 3406330, 37469344, 449632492, 5845221941, 81833107734, 1227496615330, 19639945846096, 333879079382663, 6009823428889074, 114186645148891076, 2283732902977823060, 47958390962534282489, 1055084601175754216782
Offset: 0

Views

Author

Seiichi Manyama, Jan 04 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x*sum(k=0, 2, binomial(2, k)*(-x)^k/(k+1)!)*exp(-x))/(1-x)))

Formula

a(0) = 1; a(n) = n*a(n-1) + (-1)^n * binomial(n+2,3).
a(n) = n! + (-1)^n * A368585(n).
E.g.f.: (1 - x * (1-x+x^2/6) * exp(-x)) / (1-x).

A368762 a(n) = n! * (1 + Sum_{k=0..n} binomial(k+1,2) / k!).

Original entry on oeis.org

1, 2, 7, 27, 118, 605, 3651, 25585, 204716, 1842489, 18424945, 202674461, 2432093610, 31617217021, 442641038399, 6639615576105, 106233849217816, 1805975436703025, 32507557860654621, 617643599352437989, 12352871987048759990, 259410311728023960021
Offset: 0

Views

Author

Seiichi Manyama, Jan 04 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace((1+x*sum(k=0, 1, binomial(1, k)*x^k/(k+1)!)*exp(x))/(1-x)))

Formula

a(0) = 1; a(n) = n*a(n-1) + binomial(n+1,2).
a(n) = n! + A103519(n).
E.g.f.: (1 + x * (1+x/2) * exp(x)) / (1-x).

A368764 a(n) = n! * (1 + Sum_{k=0..n} binomial(k+3,4) / k!).

Original entry on oeis.org

1, 2, 9, 42, 203, 1085, 6636, 46662, 373626, 3363129, 33632005, 369953056, 4439438037, 57712696301, 807977750594, 12119666261970, 193914660195396, 3296549223326577, 59337886019884371, 1127419834377810364, 22548396687556216135, 473516330438680549461
Offset: 0

Views

Author

Seiichi Manyama, Jan 04 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace((1+x*sum(k=0, 3, binomial(3, k)*x^k/(k+1)!)*exp(x))/(1-x)))

Formula

a(0) = 1; a(n) = n*a(n-1) + binomial(n+3,4).
a(n) = n! + A368575(n).
E.g.f.: (1 + x * (1+3*x/2+x^2/2+x^3/24) * exp(x)) / (1-x).
Showing 1-3 of 3 results.