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.

A360267 a(n) = Sum_{k=0..floor(n/4)} binomial(n-3*k,k) * binomial(2*(n-3*k),n-3*k).

Original entry on oeis.org

1, 2, 6, 20, 72, 264, 984, 3712, 14136, 54224, 209200, 810912, 3155616, 12320512, 48239232, 189336192, 744722400, 2934759360, 11584470336, 45796087680, 181285742592, 718498695424, 2850802065152, 11322567705600, 45011437903104, 179088911779328
Offset: 0

Views

Author

Seiichi Manyama, Jan 31 2023

Keywords

Comments

Diagonal of rational function 1/(1 - (x + y + x^4*y^3)). - Seiichi Manyama, Mar 23 2023

Links

Crossrefs

Cf. A006139, A360266.
Cf. A360219, A374599.

Programs

  • Mathematica
    Table[Sum[Binomial[n-3k,k]Binomial[2(n-3k),n-3k],{k,0,Floor[n/3]}],{n,0,30}] (* Harvey P. Dale, May 27 2024 *)
  • PARI
    a(n) = sum(k=0, n\4, binomial(n-3*k, k)*binomial(2*(n-3*k), n-3*k));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x*(1+x^3)))

Formula

G.f.: 1/sqrt(1 - 4*x*(1 + x^3)).
n*a(n) = 2*(2*n-1)*a(n-1) + 2*(2*n-4)*a(n-4).
a(n) ~ 1 / (2*sqrt((1 - 3*r)*Pi*n) * r^n), where r = 0.2463187933841190115229... is the positive real root of the equation -1 + 4*r + 4*r^4 = 0. - Vaclav Kotesovec, Mar 23 2023

A360219 a(n) = Sum_{k=0..floor(n/4)} (-1)^k * binomial(n-3*k,k) * binomial(2*(n-3*k),n-3*k).

Original entry on oeis.org

1, 2, 6, 20, 68, 240, 864, 3152, 11616, 43136, 161152, 604992, 2280416, 8624832, 32714688, 124399488, 474066560, 1810053120, 6922776576, 26517173760, 101710338048, 390603984896, 1501732753408, 5779500226560, 22263437981184, 85835254221824, 331193445626880
Offset: 0

Views

Author

Seiichi Manyama, Jan 31 2023

Keywords

Comments

Diagonal of rational function 1/(1 - x - y + x^4*y^3). - Seiichi Manyama, Mar 23 2023

Links

Crossrefs

Cf. A157004, A360267, A374599.

Programs

  • Maple
    A360219 := proc(n)
    add((-1)^k*binomial(n-3*k,k)*binomial(2*(n-3*k),n-3*k),k=0..n/3) ;
end proc:
seq(A360219(n),n=0..70) ; # R. J. Mathar, Mar 12 2023
  • PARI
    a(n) = sum(k=0, n\4, (-1)^k*binomial(n-3*k, k)*binomial(2*(n-3*k), n-3*k));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x*(1-x^3)))

Formula

G.f.: 1/sqrt(1 - 4*x*(1 - x^3)).
n*a(n) = 2*(2*n-1)*a(n-1) - 2*(2*n-4)*a(n-4).

A374598 Expansion of 1/sqrt(1 - 4*x - 8*x^3).

Original entry on oeis.org

1, 2, 6, 24, 94, 372, 1508, 6192, 25638, 106908, 448356, 1889040, 7989676, 33902504, 144259944, 615330784, 2630199942, 11263613484, 48315367076, 207556060816, 892819376964, 3845161246424, 16578320962104, 71548426931616, 309070048163676, 1336223562436632
Offset: 0

Views

Author

Seiichi Manyama, Jul 13 2024

Keywords

Links

Crossrefs

Cf. A157004, A360266.
Cf. A084609, A374599.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x-8*x^3))
  • PARI
    a(n) = sum(k=0, n\3, 2^k*binomial(n-2*k, k)*binomial(2*(n-2*k), n-2*k));

Formula

a(n) = Sum_{k=0..floor(n/3)} 2^k * binomial(n-2*k,k) * binomial(2*(n-2*k),n-2*k).
n*a(n) = 2*(2*n-1)*a(n-1) + 4*(2*n-3)*a(n-3).
Showing 1-3 of 3 results.

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