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 1-2 of 2 results.

A360059 a(n) = Sum_{k=0..n} (-1)^k * binomial(n+3*k+3,n-k) * Catalan(k).

Original entry on oeis.org

1, 3, 4, 3, 5, 12, 6, -13, 29, 95, -130, -304, 895, 1050, -5068, -2181, 27743, -5481, -143532, 117983, 700831, -1074414, -3163138, 7872784, 12585117, -51587107, -38040886, 312988334, 18178883, -1779688404, 1013771196, 9485832411, -11749675733, -46878057651
Offset: 0

Views

Author

Seiichi Manyama, Jan 23 2023

Keywords

Crossrefs

Cf. A360058, A360060.
Cf. A000108, A358518, A360050.

Programs

  • Mathematica
    Table[Sum[(-1)^k Binomial[n+3k+3,n-k]CatalanNumber[k],{k,0,n}],{n,0,40}] (* Harvey P. Dale, May 06 2024 *)
  • PARI
    a(n) = sum(k=0, n, (-1)^k*binomial(n+3*k+3, n-k)*binomial(2*k, k)/(k+1));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(2/((1-x)^4*(1+sqrt(1+4*x/(1-x)^4))))

Formula

a(n) = binomial(n+3,3) - Sum_{k=0..n-1} a(k) * a(n-k-1).
G.f. A(x) satisfies A(x) = 1/(1-x)^4 - x * A(x)^2.
G.f.: 2 / ( (1-x)^4 * (1 + sqrt( 1 + 4*x/(1-x)^4 )) ).
D-finite with recurrence (n+1)*a(n) +(-n-2)*a(n-1) +6*(n-2)*a(n-2) +10*(-n+2)*a(n-3) +5*(n-3)*a(n-4) +(-n+4)*a(n-5)=0. - R. J. Mathar, Jan 25 2023

A360060 a(n) = Sum_{k=0..n} (-1)^k * binomial(n+4*k+4,n-k) * Catalan(k).

Original entry on oeis.org

1, 4, 7, 5, 4, 29, 50, -83, -185, 743, 1425, -5250, -9868, 40530, 73280, -319155, -557485, 2573032, 4341065, -21107670, -34398290, 175655925, 276438452, -1479202280, -2247154681, 12581036223, 18440253397, -107916225837, -152514334540, 932452267956, 1269723550920
Offset: 0

Views

Author

Seiichi Manyama, Jan 23 2023

Keywords

Crossrefs

Cf. A360058, A360059.
Cf. A000108, A360051, A360057.

Programs

  • PARI
    a(n) = sum(k=0, n, (-1)^k*binomial(n+4*k+4, n-k)*binomial(2*k, k)/(k+1));
  • PARI
    my(N=40, 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*(-n-1)*a(n-1) +(11*n-19)*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 1-2 of 2 results.

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

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