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.

A360291 a(n) = Sum_{k=0..floor(n/3)} binomial(n-1-2*k,k) * binomial(2*n-6*k,n-3*k).

Original entry on oeis.org

1, 2, 6, 20, 72, 264, 984, 3714, 14148, 54284, 209482, 812196, 3161340, 12345658, 48348522, 189807336, 746740510, 2943359208, 11620961412, 45950375602, 181936110006, 721233025332, 2862271873966, 11370584735100, 45212101270728, 179926167512914
Offset: 0

Views

Author

Seiichi Manyama, Feb 01 2023

Keywords

Links

Crossrefs

Cf. A085362, A360290, A360292.
Cf. A360186, A360294, A383581.

Programs

  • PARI
    a(n) = sum(k=0, n\3, binomial(n-1-2*k, k)*binomial(2*n-6*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*(n-3)*a(n-3) - 2*(2*n-10)*a(n-4) - (n-6)*a(n-6).
a(n) = A383581(n) - A383581(n-3). - Seiichi Manyama, May 01 2025

A360295 a(n) = Sum_{k=0..floor(n/4)} (-1)^k * binomial(n-1-3*k,k) * binomial(2*n-8*k,n-4*k).

Original entry on oeis.org

1, 2, 6, 20, 70, 250, 912, 3372, 12590, 47362, 179230, 681528, 2601896, 9966798, 38288420, 147453664, 569092438, 2200577502, 8523612766, 33064771524, 128438624798, 499525018638, 1944918241388, 7580283784548, 29571439970136, 115459524588322, 451157870454298
Offset: 0

Views

Author

Seiichi Manyama, Feb 01 2023

Keywords

Links

Crossrefs

Cf. A360293, A360294.

Programs

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

Formula

G.f.: 1 / sqrt(1-4*x/(1+x^4)).
n*a(n) = 2*(2*n-1)*a(n-1) - 2*(n-4)*a(n-4) + 2*(2*n-13)*a(n-5) - (n-8)*a(n-8).

A360293 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(n-1-k,k) * binomial(2*n-4*k,n-2*k).

Original entry on oeis.org

1, 2, 6, 18, 58, 194, 662, 2290, 8002, 28178, 99830, 355426, 1270586, 4557682, 16396454, 59135458, 213745922, 774077986, 2808105318, 10202439858, 37118386490, 135210620194, 493082387766, 1799998114770, 6577045868866, 24052649767730, 88031695861590
Offset: 0

Views

Author

Seiichi Manyama, Feb 01 2023

Keywords

Crossrefs

Cf. A360294, A360295.

Programs

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

Formula

G.f.: 1 / sqrt(1-4*x/(1+x^2)).
n*a(n) = 2*(2*n-1)*a(n-1) - 2*(n-2)*a(n-2) + 2*(2*n-7)*a(n-3) - (n-4)*a(n-4).
a(n) ~ (1 + sqrt(3))^(2*n) / (3^(1/4) * sqrt(Pi*n) * 2^(n - 1/2)). - Vaclav Kotesovec, Feb 02 2023

A360314 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(n-1-2*k,n-3*k) * binomial(2*k,k).

Original entry on oeis.org

1, 0, 0, -2, -2, -2, 4, 10, 16, 2, -32, -86, -90, 26, 332, 646, 534, -690, -3040, -4934, -2270, 9066, 27260, 35198, 532, -101946, -232752, -230730, 158986, 1039078, 1899364, 1265370, -2714160, -9926158, -14625008, -4036358, 34062386, 89744810, 104123084
Offset: 0

Views

Author

Seiichi Manyama, Feb 03 2023

Keywords

Crossrefs

Cf. A360313, A360315.
Cf. A360294, A360309.

Programs

  • PARI
    a(n) = sum(k=0, n\3, (-1)^k*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).
Showing 1-4 of 4 results.

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