A071725 - OEIS

%I #27 Mar 23 2021 15:53:43

%S 1,1,3,10,34,117,407,1430,5070,18122,65246,236436,861764,3157325,

%T 11622015,42961470,159419670,593636670,2217608250,8308432140,

%U 31212003420,117544456770,443690433654,1678353186780,6361322162444

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

%C 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

%H Vincenzo Librandi, <a href="/A071725/b071725.txt">Table of n, a(n) for n = 0..200</a>

%H Hanna Mularczyk, <a href="https://arxiv.org/abs/1908.04025">Lattice Paths and Pattern-Avoiding Uniquely Sorted Permutations</a>, arXiv:1908.04025 [math.CO], 2019.

%F From _Paul Barry_, Jun 28 2009: (Start)

%F E.g.f.: exp(2*x)*dif(Bessel_I(1,2*x) - Bessel_I(2,2*x),x);

%F a(n) = Sum_{k=0..n} ( (-1)^k*2^(n-k)*binomial(n,k)*binomial(k+1,floor(k/2)) ). (End)

%F (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

%F a(n) ~ 3*2^(2*n+1)/(sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Mar 21 2014

%F From _G. C. Greubel_, Mar 23 2021: (Start)

%F G.f.: (1-5*x+6*x^2 - (1-3*x+2*x^2)*sqrt(1-4*x))/(2*x^3).

%F E.g.f.: exp(2*x)*(BesselI(0,2*x) -BesselI(1,2*x) +BesselI(2,2*x) -BesselI(3,2*x)).

%F a(n) = C(n+2) -3*C(n+1) +2*C(n), where C(n) are the Catalan numbers.

%F a(n) = 6*((n^2+1)/((n+2)*(n+3)))*C(n). (End)

%p A000108:= n-> binomial(2*n, n)/(n+1);

%p A071725:= n-> 6*((n^2+1)/((n+2)*(n+3)))*A000108(n);

%p seq(A071725(n), n=0..30); # _G. C. Greubel_, Mar 23 2021

%t 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 *)

%o (Magma) [6*((n^2+1)/((n+2)*(n+3)))*Catalan(n): n in [0..30]]; // _G. C. Greubel_, Mar 23 2021

%o (Sage) [6*((n^2+1)/((n+2)*(n+3)))*catalan_number(n) for n in (0..30)] # _G. C. Greubel_, Mar 23 2021

%Y Cf. A000108.

%Y Essentially the same as A026016.

%K nonn

%O 0,3

%A _N. J. A. Sloane_, Jun 06 2002