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 1-4 of 4 results.

A368574 a(n) = n! * Sum_{k=0..n} binomial(k+2,3) / k!.

Original entry on oeis.org

0, 1, 6, 28, 132, 695, 4226, 29666, 237448, 2137197, 21372190, 235094376, 2821132876, 36674727843, 513446190362, 7701692856110, 123227085698576, 2094860456876761, 37707488223782838, 716442276251875252, 14328845525037506580, 300905756025787639951, 6619926632567328080946
Offset: 0

Views

Author

Seiichi Manyama, Dec 31 2023

Keywords

Crossrefs

Programs

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

Formula

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

A368586 a(n) = n! * Sum_{k=0..n} (-1)^(n-k) * binomial(k+3,4) / k!.

Original entry on oeis.org

0, 1, 3, 6, 11, 15, 36, -42, 666, -5499, 55705, -611754, 7342413, -95449549, 1336296066, -20044437930, 320711010756, -5452087178007, 98137569210111, -1864613814984794, 37292276299704735, -783137802293788809, 17229031650463366448, -396267727960657413354
Offset: 0

Views

Author

Seiichi Manyama, Dec 31 2023

Keywords

Crossrefs

Programs

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

Formula

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

A368576 a(n) = n! * Sum_{k=0..n} binomial(k+4,5) / k!.

Original entry on oeis.org

0, 1, 8, 45, 236, 1306, 8088, 57078, 457416, 4118031, 41182312, 453008435, 5436105588, 70669378832, 989371312216, 14840569694868, 237449115133392, 4036634957288013, 72659429231210568, 1380529155393034441, 27610583107860731324, 579822245265075410934
Offset: 0

Views

Author

Seiichi Manyama, Dec 31 2023

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(x*sum(k=0, 4, binomial(4, k)*x^k/(k+1)!)*exp(x)/(1-x))))

Formula

a(0) = 0; a(n) = n*a(n-1) + binomial(n+4,5).
E.g.f.: x * (1+2*x+x^2+x^3/6+x^4/120) * 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 1-4 of 4 results.