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.

A360024 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(n-k,k) * Catalan(k).

Original entry on oeis.org

1, 1, 0, -1, 0, 3, 3, -5, -12, 5, 41, 21, -110, -165, 210, 735, -30, -2505, -2205, 6555, 13710, -10035, -57390, -18471, 185790, 240793, -436317, -1276795, 360302, 4956495, 3410749, -14776581, -26548200, 28671609, 124807175, 14211153, -446256722, -481156685
Offset: 0

Views

Author

Seiichi Manyama, Jan 22 2023

Keywords

Links

Crossrefs

Cf. A360025, A360026, A360027.
Cf. A000108.

Programs

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

Formula

a(n) = 1 - Sum_{k=0..n-2} a(k) * a(n-k-2).
G.f. A(x) satisfies: A(x) = 1/(1-x) - x^2 * A(x)^2.
G.f.: 2 / ( 1-x + sqrt((1-x)^2 + 4*x^2*(1-x)) ).
D-finite with recurrence (n+2)*a(n) +2*(-n-1)*a(n-1) +(5*n-4)*a(n-2) +2*(-2*n+3)*a(n-3)=0. - R. J. Mathar, Jan 25 2023

A360025 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(n-2*k,k) * Catalan(k).

Original entry on oeis.org

1, 1, 1, 0, -1, -2, -1, 2, 7, 9, 3, -16, -39, -43, 9, 126, 247, 199, -213, -984, -1555, -756, 2525, 7518, 9593, 559, -24899, -56216, -55241, 33150, 225879, 407194, 273199, -529745, -1938549, -2822128, -833219, 6083986, 15904733, 18288966, -4172187, -61154333
Offset: 0

Views

Author

Seiichi Manyama, Jan 22 2023

Keywords

Links

Crossrefs

Cf. A360024, A360026, A360027.
Cf. A000108, A216604, A349047.

Programs

  • PARI
    a(n) = sum(k=0, n\3, (-1)^k*binomial(n-2*k, k)*binomial(2*k, k)/(k+1));
  • PARI
    my(N=50, x='x+O('x^N)); Vec(2/(1-x+sqrt((1-x)^2+4*x^3*(1-x))))

Formula

a(n) = 1 - Sum_{k=0..n-3} a(k) * a(n-k-3).
G.f. A(x) satisfies: A(x) = 1/(1-x) - x^3 * A(x)^2.
G.f.: 2 / ( 1-x + sqrt((1-x)^2 + 4*x^3*(1-x)) ).
D-finite with recurrence +(n+3)*a(n) +2*(-n-2)*a(n-1) +(n+1)*a(n-2) +2*(2*n-3)*a(n-3) +4*(-n+2)*a(n-4)=0. - R. J. Mathar, Jan 25 2023

A360026 a(n) = Sum_{k=0..floor(n/4)} (-1)^k * binomial(n-3*k,k) * Catalan(k).

Original entry on oeis.org

1, 1, 1, 1, 0, -1, -2, -3, -2, 1, 6, 13, 17, 13, -4, -39, -83, -113, -92, 31, 279, 605, 850, 701, -219, -2129, -4736, -6749, -5690, 1569, 17114, 38713, 55957, 48249, -11498, -142163, -326860, -478957, -421262, 84015, 1210831, 2829363, 4197670, 3762583, -601732
Offset: 0

Views

Author

Seiichi Manyama, Jan 22 2023

Keywords

Links

Crossrefs

Cf. A360024, A360025, A360027.
Cf. A000108, A346073, A349048.

Programs

  • PARI
    a(n) = sum(k=0, n\4, (-1)^k*binomial(n-3*k, k)*binomial(2*k, k)/(k+1));
  • PARI
    my(N=50, x='x+O('x^N)); Vec(2/(1-x+sqrt((1-x)^2+4*x^4*(1-x))))

Formula

a(n) = 1 - Sum_{k=0..n-4} a(k) * a(n-k-4).
G.f. A(x) satisfies: A(x) = 1/(1-x) - x^4 * A(x)^2.
G.f.: 2 / ( 1-x + sqrt((1-x)^2 + 4*x^4*(1-x)) ).
D-finite with recurrence +(n+4)*a(n) +2*(-n-3)*a(n-1) +(n+2)*a(n-2) +4*(n-2)*a(n-4) +2*(-2*n+5)*a(n-5)=0. - R. J. Mathar, Jan 25 2023
Showing 1-3 of 3 results.

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