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.

A068768 Generalized Catalan numbers 6*x*A(x)^2 -A(x) +1 -5*x =0.

Original entry on oeis.org

1, 1, 12, 150, 1944, 25992, 356832, 5008824, 71629920, 1040509152, 15315578496, 227981324736, 3426473187072, 51929043390720, 792725911280640, 12178706839758720, 188158789025809920, 2921622674591946240
Offset: 0

Views

Author

Wolfdieter Lang, Mar 04 2002

Keywords

Comments

a(n) = K(6,6; n)/6 with K(a,b; n) defined in a comment to A068763.

Links

Crossrefs

Cf. A000108, A068764-A068767, A068769-A068772, A025227-A025230.

Programs

  • Mathematica
    CoefficientList[Series[(1-Sqrt[1-24*x*(1-5*x)])/(12*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 04 2014 *)

Formula

a(n) = (6^n) * p(n, -5/6) with the row polynomials p(n, x) defined from array A068763.
a(n+1) = 6*sum(a(k)*a(n-k), k=0..n), n>=1, a(0)=1=a(1).
G.f.: (1-sqrt(1-24*x*(1-5*x)))/(12*x).
D-finite with recurrence: (n+1)*a(n) = 120*(2-n)*a(n-2) + 12*(2*n-1)*a(n-1). - Fung Lam, Mar 04 2014
a(n) ~ sqrt(3+3*sqrt(6)) * (12+2*sqrt(6))^n / (6*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 04 2014

A068770 Generalized Catalan numbers 8*x*A(x)^2 -A(x) +1 -7*x=0.

Original entry on oeis.org

1, 1, 16, 264, 4480, 77952, 1386496, 25135616, 463233024, 8658673664, 163829383168, 3132565553152, 60446638866432, 1175715287400448, 23028562592268288, 453848132868898816, 8993594212565909504
Offset: 0

Views

Author

Wolfdieter Lang, Mar 04 2002

Keywords

Comments

a(n) = K(8,8; n)/8 with K(a,b; n) defined in a comment to A068763.

Links

Crossrefs

Cf. A000108, A068764-A068769, A068771-A068772, A025227-A025230.

Programs

  • Mathematica
    CoefficientList[Series[(1-Sqrt[1-32*x*(1-7*x)])/(16*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 04 2014 *)
  • PARI
    a(n) = if(n, (4^(n-1)*14^(1/2*n+1/2)*pollegendre(n+1,2/7*14^(1/2)) - pollegendre(n,2/7*14^(1/2))*4^n*14^(n/2))\/n, 1) \\ Charles R Greathouse IV, Mar 19 2017

Formula

a(n) = (8^n) * p(n, -7/8) with the row polynomials p(n, x) defined from array A068763.
a(n+1) = 8*sum(a(k)*a(n-k), k=0..n), n>=1, a(0)=1=a(1).
G.f.: (1-sqrt(1-32*x*(1-7*x)))/(16*x).
a(n) = (4^(n-1)*14^(1/2*n+1/2)*LegendreP(n+1,2/7*14^(1/2)) - LegendreP(n,2/7*14^(1/2))*4^n*14^(1/2*n))/n for n > 0. - Mark van Hoeij, Apr 23 2010
Recurrence: (n+1)*a(n) = 224*(2-n)*a(n-2) + 16*(2*n-1)*a(n-1). - Fung Lam, Mar 04 2014
a(n) ~ sqrt(1+2*sqrt(2)) * (16+4*sqrt(2))^n / (4*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 04 2014
Showing 1-2 of 2 results.

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

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