OFFSET
0,3
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
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Hanna Mularczyk, Lattice Paths and Pattern-Avoiding Uniquely Sorted Permutations, arXiv:1908.04025 [math.CO], 2019.
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)
MAPLE
A000108:= n-> binomial(2*n, n)/(n+1);
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 *)
PROG
(Magma) [6*((n^2+1)/((n+2)*(n+3)))*Catalan(n): n in [0..30]]; // G. C. Greubel, Mar 23 2021
(Sage) [6*((n^2+1)/((n+2)*(n+3)))*catalan_number(n) for n in (0..30)] # G. C. Greubel, Mar 23 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jun 06 2002
STATUS
approved