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

0,4

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

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

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 *)

Prog

(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
(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

Crossrefs

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

Sequence in context: A206960 A145543 A141043 * A192876 A201689 A356650

Adjacent sequences: A111157 A111158 A111159 * A111161 A111162 A111163

Keyword

nonn

Author

N. J. A. Sloane, Oct 22 2005

Status

approved