A111160 G.f.: C - Z; where C is the g.f. for the Catalan numbers (A000108) and Z is the g.f. for A055113 with offset 0.

0, 1, 1, 4, 9, 31, 91, 309, 1009, 3481, 11956, 42065, 148655, 532039, 1915369, 6950452, 25357233, 93034813, 342888250, 1269246437, 4715945712, 17583623988, 65766726906, 246694006971, 927801717255, 3497918129001, 13217196871126, 50046561077947

Offset: 0

Author

N. J. A. Sloane, Oct 22 2005

Keywords

• nonn

Comments

Expressible in terms of ballot numbers.

Number of positive walks with n steps {-2,-1,1,2} starting at the origin, ending at altitude 2, and staying strictly above the x-axis. - David Nguyen, Dec 16 2016

Links

• T. D. Noe, Table of n, a(n) for n=0..200
• C. Banderier, C. Krattenthaler, A. Krinik, D. Kruchinin, V. Kruchinin, D. Nguyen, and M. Wallner, Explicit formulas for enumeration of lattice paths: basketball and the kernel method, arXiv preprint arXiv:1609.06473 [math.CO], 2016.
• Jean-Luc Baril and José L. Ramírez, Knight's paths towards Catalan numbers, Univ. Bourgogne Franche-Comté (2022).
• D. G. Rogers, Comments on A111160, A055113 and A006013

Crossrefs

Cf. A000108, A055113, A006013, A187430, A276901, A306668.

Programs

Magma

I:=[1,1,4]; [0] cat [n le 3 select I[n] else (n*(115*n^3 - 344*n^2 + 299*n - 82)*Self(n-1) + 4*(2*n-3)*(5*n^3 + 27*n^2 - 74*n + 30)*Self(n-2) - 36*(n-2)*(2*n-5)*(2*n-3)*(5*n-3)*Self(n-3))/(2*n*(n+1)*(2*n+1)*(5*n-8)): n in [1..30]]; // Vincenzo Librandi, Oct 06 2015

Maple

a := n -> (-1)^(n+1)*binomial(2*n+1,n)*hypergeom([-n-1,n/2+1/2,n/2],[n,n+1],4)/ (2*n+1);
[0, op([seq(round(evalf(a(n),32)), n=1..27)])]; # Peter Luschny, Oct 06 2015

Mathematica

CoefficientList[ Series[ -((-3 + Sqrt[1 - 4*x] + Sqrt[2]*Sqrt[1 + Sqrt[1 - 4x] + 6x])/(4x)), {x, 0, 10}], x] (* Robert G. Wilson v *)

PARI

a(n) = if(n==0, 0, sum(k=0, (n+1)/2, binomial(n-k,n-2*k+1)*binomial(2*n+1,k))/(2*n+1)); \\ Altug Alkan, Oct 05 2015

Formula

Let C := (1 - sqrt(1 - 4*x)) / (2*x), Z := (- 1/4 - (1/4)*(1 - 4*x)^(1/2) + (1/4)*(2 + 2*(1 - 4*x)^(1/2) + 12*x)^(1/2))/x; g.f. is W := C - Z.

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

a(n) = sum(j=0..n+1, binomial(n+2*j-1,j)*(-1)^(n+j+1)*binomial(2*n+1,j+n))/(2*n+1). [Vladimir Kruchinin, Feb 15 2013]

a(n) ~ (1+1/sqrt(5))*2^(2*n-1)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 13 2013

Recurrence: 2*n*(n+1)*(2*n+1)*(5*n-8)*a(n) = n*(115*n^3 - 344*n^2 + 299*n - 82)*a(n-1) + 4*(2*n-3)*(5*n^3 + 27*n^2 - 74*n + 30)*a(n-2) - 36*(n-2)*(2*n-5)*(2*n-3)*(5*n-3)*a(n-3). - Vaclav Kotesovec, Aug 13 2013

a(n) = Sum_{j=0..(n+1)/2}(binomial(n-j,n-2*j+1)*binomial(2*n+1,j))/(2*n+1). - Vladimir Kruchinin, Oct 05 2015

a(n) = (-1)^(n+1)*C(2*n+1,n)*hypergeom([-n-1,n/2+1/2,n/2],[n,n+1],4)/(2*n+1) for n>0. - Peter Luschny, Oct 06 2015