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

A360027 a(n) = Sum_{k=0..floor(n/5)} (-1)^k * binomial(n-4*k,k) * Catalan(k).

Original entry on oeis.org

1, 1, 1, 1, 1, 0, -1, -2, -3, -4, -3, 0, 5, 12, 21, 27, 25, 10, -23, -79, -149, -210, -225, -143, 101, 544, 1153, 1783, 2135, 1714, -81, -3735, -9263, -15724, -20603, -19490, -6485, 24242, 75307, 140955, 200891, 215530, 126527, -132122, -605687
Offset: 0

Views

Author

Seiichi Manyama, Jan 22 2023

Keywords

Links

Crossrefs

Cf. A360024, A360025, A360026.
Cf. A000108, A346074.

Programs

  • PARI
    a(n) = sum(k=0, n\5, (-1)^k*binomial(n-4*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^5*(1-x))))

Formula

a(n) = 1 - Sum_{k=0..n-5} a(k) * a(n-k-5).
G.f. A(x) satisfies: A(x) = 1/(1-x) - x^5 * A(x)^2.
G.f.: 2 / ( 1-x + sqrt((1-x)^2 + 4*x^5*(1-x)) ).
D-finite with recurrence (n+5)*a(n) 2*(-n-4)*a(n-1) +(n+3)*a(n-2) +2*(2*n-5)*a(n-5) +4*(-n+3)*a(n-6)=0. - R. J. Mathar, Jan 25 2023
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.