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.

A386362 Expansion of (1/x) * Series_Reversion( x/(1+7*x+9*x^2) ).

Original entry on oeis.org

1, 7, 58, 532, 5209, 53347, 564499, 6123481, 67732483, 761052565, 8662502212, 99671232514, 1157409133831, 13546774268125, 159649564550746, 1892849564159596, 22562032457415067, 270209749616920813, 3249905798884688038, 39237866746912398292, 475388228365424562019
Offset: 0

Views

Author

Seiichi Manyama, Aug 20 2025

Keywords

Crossrefs

Column k=3 of A386408.
Cf. A002212, A127846, A386389.
Cf. A000108, A337167, A385716.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1+7*x+9*x^2))/x)

Formula

G.f.: 2/(1 - 7*x + sqrt((1-x) * (1-13*x))).
a(n) = (A337167(n+1) - A337167(n))/3.
(n+2)*a(n) = 7*(2*n+1)*a(n-1) - 13*(n-1)*a(n-2) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 9^k * 7^(n-2*k) * binomial(n,2*k) * Catalan(k).
a(n) = Sum_{k=0..n} 3^k * binomial(n,k) * Catalan(k+1).

A386389 Expansion of (1/x) * Series_Reversion( x/(1+9*x+16*x^2) ).

Original entry on oeis.org

1, 9, 97, 1161, 14849, 198729, 2748641, 38977353, 563644673, 8280210825, 123226850913, 1853870946057, 28148395838721, 430791367720905, 6638484468424929, 102918165951351753, 1604104541561284097, 25121009971212463881, 395085505395126968417, 6237523016309454855561
Offset: 0

Views

Author

Seiichi Manyama, Aug 20 2025

Keywords

Crossrefs

Column k=4 of A386408.
Cf. A002212, A127846, A386362.
Cf. A000108, A269796, A386387.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1+9*x+16*x^2))/x)

Formula

G.f.: 2/(1 - 9*x + sqrt((1-x) * (1-17*x))).
a(n) = (A386387(n+1) - A386387(n))/4.
(n+2)*a(n) = 9*(2*n+1)*a(n-1) - 17*(n-1)*a(n-2) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 16^k * 9^(n-2*k) * binomial(n,2*k) * Catalan(k).
a(n) = Sum_{k=0..n} 4^k * binomial(n,k) * Catalan(k+1).

A386432 a(n) = Sum_{k=0..n} n^k * binomial(n,k) * Catalan(k+1).

Original entry on oeis.org

1, 3, 29, 532, 14849, 562551, 27053749, 1581258225, 108965790593, 8657148898585, 779508506302701, 78480330282178738, 8738801236865140417, 1066555304017996550265, 141604665239501105707269, 20321162053065050407161076, 3134730687100285268294654465, 517309567362171441488395248225
Offset: 0

Views

Author

Seiichi Manyama, Aug 20 2025

Keywords

Links

Crossrefs

Main diagonal of A386408.
Cf. A000108, A338979.

Programs

  • Magma
    [&+[n^k*Binomial(n, k) * Catalan (k+1): k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Aug 22 2025
  • Mathematica
    Table[Sum[(n^k/. 0^0->1)*Binomial[n,k]*CatalanNumber[k+1],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 22 2025 *)
  • PARI
    a(n) = sum(k=0, n, n^k*binomial(n, k)*(2*(k+1))!/((k+1)!*(k+2)!));

Formula

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

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