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.

A360045 a(n) = Sum_{k=0..floor(n/3)} binomial(n+2,3*k+2) * Catalan(k).

Original entry on oeis.org

1, 3, 6, 11, 21, 42, 86, 180, 387, 852, 1908, 4329, 9933, 23013, 53748, 126386, 298953, 710847, 1698086, 4073181, 9806565, 23689555, 57401322, 139475190, 339767545, 829638396, 2030206248, 4978136133, 12229451631, 30095772966, 74184390468, 183139941384
Offset: 0

Views

Author

Seiichi Manyama, Jan 23 2023

Keywords

Crossrefs

Cf. A086615, A360046, A360047.
Cf. A000108.

Programs

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

Formula

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

A360046 a(n) = Sum_{k=0..floor(n/4)} binomial(n+3,4*k+3) * Catalan(k).

Original entry on oeis.org

1, 4, 10, 20, 36, 64, 120, 240, 497, 1036, 2158, 4524, 9625, 20816, 45560, 100368, 221915, 492380, 1097302, 2457228, 5526666, 12474000, 28233600, 64061920, 145704327, 332174532, 758977386, 1737703780, 3985847284, 9157908736, 21074460512, 48569746368, 112096071675
Offset: 0

Views

Author

Seiichi Manyama, Jan 23 2023

Keywords

Links

Crossrefs

Cf. A086615, A360045, A360047.
Cf. A000108.

Programs

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

Formula

a(n) = binomial(n+3,3) + Sum_{k=0..n-4} a(k) * a(n-k-4).
G.f. A(x) satisfies: A(x) = 1/(1-x)^4 + x^4 * A(x)^2.
G.f.: 2 / ( (1-x)^2 * ((1-x)^2 + sqrt((1-x)^4 - 4*x^4)) ).
D-finite with recurrence (n+4)*a(n) +5*(-n-3)*a(n-1) +10*(n+2)*a(n-2) +10*(-n-1)*a(n-3) +(n+8)*a(n-4) +3*(n-1)*a(n-5)=0. - R. J. Mathar, Jan 25 2023

A360057 a(n) = Sum_{k=0..n} binomial(n+4*k+4,n-k) * Catalan(k).

Original entry on oeis.org

1, 6, 27, 125, 644, 3643, 21974, 138395, 898695, 5970927, 40386209, 277127148, 1924349756, 13496536510, 95467320600, 680260392219, 4878382821267, 35182209381590, 255000022472565, 1856501085686340, 13570366067586294, 99554601986349471, 732756800760507312
Offset: 0

Views

Author

Seiichi Manyama, Jan 23 2023

Keywords

Crossrefs

Cf. A086616, A162481, A358518.
Cf. A000108, A360047.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n+4*k+4, n-k)*binomial(2*k, k)/(k+1));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(2/((1-x)^5*(1+sqrt(1-4*x/(1-x)^5))))

Formula

a(n) = binomial(n+4,4) + Sum_{k=0..n-1} a(k) * a(n-k-1).
G.f. A(x) satisfies A(x) = 1/(1-x)^5 + x * A(x)^2.
G.f.: 2 / ( (1-x)^5 * (1 + sqrt( 1 - 4*x/(1-x)^5 )) ).
D-finite with recurrence (n+1)*a(n) +2*(-5*n+1)*a(n-1) +(19*n-11)*a(n-2) +20*(-n+2)*a(n-3) +15*(n-3)*a(n-4) +6*(-n+4)*a(n-5) +(n-5)*a(n-6)=0. - R. J. Mathar, Jan 25 2023
Showing 1-3 of 3 results.

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