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.

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

0,3

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);
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 *)

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

Cf. A000108.

Essentially the same as A026016.

Sequence in context: A059738 A094832 A217778 * A026016 A109263 A136439

Adjacent sequences: A071722 A071723 A071724 * A071726 A071727 A071728

Keyword

nonn

Author

N. J. A. Sloane, Jun 06 2002

Status

approved