From N. J. A. Sloane, Jul 14 2009: (Start)
The following remarks and formulas are basically copied from the ApagoduZeilberger reference, where this sequence appears as an example.
These are the (oldtime) 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 basketballscoring 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 thirdorder 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
