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.

A360309 a(n) = Sum_{k=0..floor(n/3)} binomial(n-1-2*k,n-3*k) * binomial(2*k,k).

Original entry on oeis.org

1, 0, 0, 2, 2, 2, 8, 14, 20, 46, 92, 158, 314, 630, 1176, 2274, 4498, 8674, 16804, 32990, 64358, 125414, 245832, 481674, 942912, 1850122, 3633220, 7133730, 14020694, 27578954, 54261912, 106819006, 210411028, 414619486, 817344908, 1611978734, 3180333830, 6276743430
Offset: 0

Views

Author

Seiichi Manyama, Feb 03 2023

Keywords

Crossrefs

Cf. A026585, A360310.
Cf. A360291, A360314.

Programs

  • PARI
    a(n) = sum(k=0, n\3, binomial(n-1-2*k, n-3*k)*binomial(2*k, k));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(1/sqrt(1-4*x^3/(1-x)))

Formula

G.f.: 1 / sqrt(1-4*x^3/(1-x)).
n*a(n) = 2*(n-1)*a(n-1) - (n-2)*a(n-2) + 2*(2*n-3)*a(n-3) - 2*(2*n-6)*a(n-4).
a(n) ~ 2^(n-1) / sqrt(Pi*n). - Vaclav Kotesovec, Feb 18 2023

A360313 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(n-1-k,n-2*k) * binomial(2*k,k).

Original entry on oeis.org

1, 0, -2, -2, 4, 10, -4, -38, -22, 114, 188, -234, -914, -18, 3376, 3338, -9416, -21718, 14416, 96338, 39274, -328558, -471344, 795398, 2586064, -517690, -10453424, -8272658, 32186818, 63596494, -61876584, -307070174, -62655330, 1129250706, 1356328788
Offset: 0

Views

Author

Seiichi Manyama, Feb 03 2023

Keywords

Links

Crossrefs

Cf. A360314, A360315.
Cf. A026585, A360293.

Programs

  • Mathematica
    RecurrenceTable[{a[0]==1,a[1]==0,a[2]==-2,a[n]==1/n (2(n-1)a[n-1]-(5n-6)a[n-2]+2(2n-5)a[n-3])},a,{n,40}] (* Harvey P. Dale, Sep 20 2024 *)
  • PARI
    a(n) = sum(k=0, n\2, (-1)^k*binomial(n-1-k, n-2*k)*binomial(2*k, k));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(1/sqrt(1+4*x^2/(1-x)))

Formula

G.f.: 1 / sqrt(1+4*x^2/(1-x)).
n*a(n) = 2*(n-1)*a(n-1) - (5*n-6)*a(n-2) + 2*(2*n-5)*a(n-3).

A360315 a(n) = Sum_{k=0..floor(n/4)} (-1)^k * binomial(n-1-3*k,n-4*k) * binomial(2*k,k).

Original entry on oeis.org

1, 0, 0, 0, -2, -2, -2, -2, 4, 10, 16, 22, 8, -26, -80, -154, -178, -82, 204, 750, 1374, 1642, 868, -1886, -6886, -12802, -15784, -8538, 17166, 64554, 122476, 152602, 86056, -157642, -616456, -1183666, -1493402, -878250, 1468080
Offset: 0

Views

Author

Seiichi Manyama, Feb 03 2023

Keywords

Crossrefs

Cf. A360313, A360314.
Cf. A360295, A360310.

Programs

  • PARI
    a(n) = sum(k=0, n\4, (-1)^k*binomial(n-1-3*k, n-4*k)*binomial(2*k, k));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(1/sqrt(1+4*x^4/(1-x)))

Formula

G.f.: 1 / sqrt(1+4*x^4/(1-x)).
n*a(n) = 2*(n-1)*a(n-1) - (n-2)*a(n-2) - 2*(2*n-4)*a(n-4) + 2*(2*n-7)*a(n-5).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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