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%I #44 Mar 28 2021 22:15:56
%S 1,2,14,84,556,3736,25612,177688,1244398,8777612,62271384,443847648,
%T 3175924636,22799963576,164142004184,1184574592592,8567000931404,
%U 62073936511496,450518481039956,3274628801768744,23833760489660324
%N T(2n,n) with T as in A036355.
%C From _N. J. A. Sloane_, Jul 14 2009: (Start)
%C The following remarks and formulas are basically copied from the Apagodu-Zeilberger reference, where this sequence appears as an example.
%C 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.
%C 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)}.
%C It satisfies the third-order linear recurrence:
%C (16/5)(2n + 3)(11n + 26)(1 + n)/((n + 3)(2 + n)(11n + 15))a(n)
%C -(4/5)(121n^3 + 649n^2 + 1135n + 646)/((n + 3)(2 + n)(11n + 15))a(1 + n)
%C -(2/5)(176n^2 + 680n + 605)/((11n + 15)(n + 3))a(2 + n) + a(n + 3) = 0 ;
%C subject to the initial conditions: a(0) = 1; a(1) = 2; a(2) = 14 :
%C 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).
%C (End)
%C In closed form, multiplicative constant is sqrt((15+8*sqrt(3))/(66*Pi)) = 0.37305616313160230... - _Vaclav Kotesovec_, Oct 24 2012
%C Diagonal of rational function 1/(1 - (x + y + x^2 + y^2)). - _Gheorghe Coserea_, Aug 06 2018
%H Reinhard Zumkeller, <a href="/A036692/b036692.txt">Table of n, a(n) for n = 0..1000</a>
%H Moa Apagodu and Doron Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/appswz.html">FIVE Applications of Wilf-Zeilberger Theory to Enumeration and Probability</a>; <a href="/A036692/a036692.pdf">Local copy</a> [Pdf file only, no active links]
%F 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
%t 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 *)
%o (PARI) /* same as in A092566 but use */
%o steps=[[1,0], [2,0], [0,1], [0,2]];
%o /* _Joerg Arndt_, Jun 30 2011 */
%o (Haskell)
%o a036692 n = a036355 (2 * n) n -- _Reinhard Zumkeller_, Apr 24 2013
%Y Cf. A000984, A036355.
%K nonn
%O 0,2
%A _Floor van Lamoen_
%E Extended by _Christian G. Bower_, Nov 18 2003
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