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.

A369723 Expansion of e.g.f. exp( 2 * (1-sqrt(1-4*x)) ).

Original entry on oeis.org

1, 4, 24, 208, 2464, 37824, 720256, 16450816, 439245312, 13440572416, 464007387136, 17847329869824, 757011972726784, 35108108023349248, 1767517592731090944, 96007679735852498944, 5596725706163142393856, 348533116657888468402176
Offset: 0

Views

Author

Seiichi Manyama, Jan 30 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(2*(1-sqrt(1-4*x)))))

Formula

a(0) = 1; a(n) = Sum_{k=0..n-1} 4^(n-k) * (n-1+k)! / (k! * (n-1-k)!).
a(n) = 2*(2*n-3)*a(n-1) + 16*a(n-2).

A369724 Expansion of e.g.f. exp( (5/2) * (1-sqrt(1-4*x)) ).

Original entry on oeis.org

1, 5, 35, 335, 4225, 67525, 1321075, 30751775, 832573025, 25745985125, 896177819875, 34698406783375, 1479737530398625, 68935386567921125, 3483762766656021875, 189846574063623209375, 11098195364856546690625, 692834276972696475053125
Offset: 0

Views

Author

Seiichi Manyama, Jan 30 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(5/2*(1-sqrt(1-4*x)))))

Formula

a(0) = 1; a(n) = Sum_{k=0..n-1} 5^(n-k) * (n-1+k)! / (k! * (n-1-k)!).
a(n) = 2*(2*n-3)*a(n-1) + 25*a(n-2).

A369746 Expansion of e.g.f. exp( 3 * (1-sqrt(1-2*x)) ).

Original entry on oeis.org

1, 3, 12, 63, 423, 3528, 35559, 422901, 5817744, 91072269, 1600588269, 31230827532, 670252672593, 15696888917427, 398454496989012, 10899543418960167, 319672849622745951, 10007954229075765984, 333139545206104991031, 11749955670275356579941
Offset: 0

Views

Author

Seiichi Manyama, Jan 30 2024

Keywords

Crossrefs

Programs

  • Maple
    # The row polynomials of A132062 evaluated at x = 3.
    T := proc(n, k) option remember; if k = 0 then 0^n elif n < k then 0
    else (2*(n - 1) - k)*T(n - 1, k) + T(n - 1, k - 1) fi end:
    seq(add(T(n, k)*3^k, k = 0..n), n = 0..19);  # Peter Luschny, Apr 25 2024
  • Mathematica
    With[{nn=20},CoefficientList[Series[Exp[3(1-Sqrt[1-2x])],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 14 2025 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(3*(1-sqrt(1-2*x)))))

Formula

a(0) = 1; a(n) = Sum_{k=0..n-1} 3^(n-k) * (n-1+k)! / (2^k * k! * (n-1-k)!).
a(n) = (2*n-3)*a(n-1) + 9*a(n-2).
Showing 1-3 of 3 results.