A036692 T(2n,n) with T as in A036355.

1, 2, 14, 84, 556, 3736, 25612, 177688, 1244398, 8777612, 62271384, 443847648, 3175924636, 22799963576, 164142004184, 1184574592592, 8567000931404, 62073936511496, 450518481039956, 3274628801768744, 23833760489660324

Offset: 0

Author

Floor van Lamoen

Keywords

• nonn

Comments

From N. J. A. Sloane, Jul 14 2009: (Start)

The following remarks and formulas are basically copied from the Apagodu-Zeilberger reference, where this sequence appears as an example.

These are the (old-time) basketball numbers, giving the number of ways a basketball game that ended with the score n : n can proceed. Recall that in the old days (before 1961), an atom of basketball-scoring could be only of one or two points.

Equivalently, this number is the number of ways of walking, in the square lattice, from (0; 0) to (n; n) using the atomic steps {(1; 0); (2; 0); (0; 1); (0; 2)}.

It satisfies the third-order linear recurrence:

(16/5)(2n + 3)(11n + 26)(1 + n)/((n + 3)(2 + n)(11n + 15))a(n)

-(4/5)(121n^3 + 649n^2 + 1135n + 646)/((n + 3)(2 + n)(11n + 15))a(1 + n)

-(2/5)(176n^2 + 680n + 605)/((11n + 15)(n + 3))a(2 + n) + a(n + 3) = 0 ;

subject to the initial conditions: a(0) = 1; a(1) = 2; a(2) = 14 :

Asymptotics: (0.37305616)(4 + 2*sqrt(3))^n*n^(-1/2)(1 + (67/1452)*sqrt(3) -(119/484))/n +((6253/117128) -(7163/234256)sqrt(3))/n^2 +(-(32645/ 15460896) sqrt(3) +(129625/10307264))/n^3).

(End)

In closed form, multiplicative constant is sqrt((15+8*sqrt(3))/(66*Pi)) = 0.37305616313160230... - Vaclav Kotesovec, Oct 24 2012

Diagonal of rational function 1/(1 - (x + y + x^2 + y^2)). - Gheorghe Coserea, Aug 06 2018

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
• Moa Apagodu and Doron Zeilberger, FIVE Applications of Wilf-Zeilberger Theory to Enumeration and Probability; Local copy [Pdf file only, no active links]

Crossrefs

Cf. A000984, A036355.

Programs

Haskell

a036692 n = a036355 (2 * n) n  -- Reinhard Zumkeller, Apr 24 2013

Mathematica

CoefficientList[Series[((3-4*x+2*(4*x^2-8*x+1)^(1/2))/((8*x+5)*(4*x^2-8*x+1)))^(1/2), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 24 2012 *)

PARI

/* same as in A092566 but use */
steps=[[1,0], [2,0], [0,1], [0,2]];
/* Joerg Arndt, Jun 30 2011 */

Formula

G.f.: ((3-4*x+2*(4*x^2-8*x+1)^(1/2))/((8*x+5)*(4*x^2-8*x+1)))^(1/2). - Mark van Hoeij, Oct 30 2011

Extensions

Extended by Christian G. Bower, Nov 18 2003