| Comment 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 (moused from pdf file, should be checked!): (.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)
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