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.

A368768 a(n) = n! * (1 + Sum_{k=0..n} (-1)^k * binomial(k+3,4) / k!).

Original entry on oeis.org

1, 0, 5, 0, 35, 105, 756, 5082, 40986, 368379, 3684505, 40528554, 486344013, 6322470349, 88514587266, 1327718805930, 21243500898756, 361139515274007, 6500511274938111, 123509714223816794, 2470194284476344735, 51874079974003228809, 1141229759428071046448
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) + (-1)^n * binomial(n+3,4).
a(n) = n! + (-1)^n * A368586(n).
E.g.f.: (1 - x * (1-3*x/2+x^2/2-x^3/24) * 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).

A368763 a(n) = n! * (1 + Sum_{k=0..n} binomial(k+2,3) / k!).

Original entry on oeis.org

1, 2, 8, 34, 156, 815, 4946, 34706, 277768, 2500077, 25000990, 275011176, 3300134476, 42901748643, 600624481562, 9009367224110, 144149875586576, 2450547884972761, 44109861929510838, 838087376660707252, 16761747533214146580, 351996698197497079951
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) + binomial(n+2,3).
a(n) = n! + A368574(n).
E.g.f.: (1 + x * (1+x+x^2/6) * exp(x)) / (1-x).
Showing 1-3 of 3 results.