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.

A071725 Expansion of (1+x^2*C^4)*C, where C = (1 - sqrt(1-4*x))/(2*x) is g.f. for Catalan numbers, A000108.

Original entry on oeis.org

1, 1, 3, 10, 34, 117, 407, 1430, 5070, 18122, 65246, 236436, 861764, 3157325, 11622015, 42961470, 159419670, 593636670, 2217608250, 8308432140, 31212003420, 117544456770, 443690433654, 1678353186780, 6361322162444
Offset: 0

Views

Author

N. J. A. Sloane, Jun 06 2002

Keywords

Comments

a(n) is the number of Dyck (n+3)-paths for which the first downstep followed by an upstep (or by nothing at all) is in position 6. For example, a(2)=3 counts UUUUDdUDDD, UUUDDdUUDD, UUUDDdUDUD (the downstep in position 6 is in small type). - David Callan, Dec 09 2004

Crossrefs

Cf. A000108.
Essentially the same as A026016.

Programs

  • Magma
    [6*((n^2+1)/((n+2)*(n+3)))*Catalan(n): n in [0..30]]; // G. C. Greubel, Mar 23 2021
    
  • Maple
    A000108:= n-> binomial(2*n, n)/(n+1);
    A071725:= n-> 6*((n^2+1)/((n+2)*(n+3)))*A000108(n);
    seq(A071725(n), n=0..30); # G. C. Greubel, Mar 23 2021
  • Mathematica
    CoefficientList[Series[(1 +x^2((1-Sqrt[1-4x])/(2x))^4)(1-Sqrt[1-4x])/(2x), {x, 0, 30}], x] (* Vaclav Kotesovec, Mar 21 2014 *)
  • Sage
    [6*((n^2+1)/((n+2)*(n+3)))*catalan_number(n) for n in (0..30)] # G. C. Greubel, Mar 23 2021

Formula

From Paul Barry, Jun 28 2009: (Start)
E.g.f.: exp(2*x)*dif(Bessel_I(1,2*x) - Bessel_I(2,2*x),x);
a(n) = Sum_{k=0..n} ( (-1)^k*2^(n-k)*binomial(n,k)*binomial(k+1,floor(k/2)) ). (End)
(n+31)*(n+3)*a(n) +(n^2-180*n-219)*a(n-1) -10*(2*n-3)*(n-10)*a(n-2) = 0. - R. J. Mathar, Nov 23 2011
a(n) ~ 3*2^(2*n+1)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 21 2014
From G. C. Greubel, Mar 23 2021: (Start)
G.f.: (1-5*x+6*x^2 - (1-3*x+2*x^2)*sqrt(1-4*x))/(2*x^3).
E.g.f.: exp(2*x)*(BesselI(0,2*x) -BesselI(1,2*x) +BesselI(2,2*x) -BesselI(3,2*x)).
a(n) = C(n+2) -3*C(n+1) +2*C(n), where C(n) are the Catalan numbers.
a(n) = 6*((n^2+1)/((n+2)*(n+3)))*C(n). (End)