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%I #28 Sep 08 2022 08:44:59
%S 2,2,3,3,5,12,37,128,464,1718,6437,24312,92380,352718,1352080,5200302,
%T 20058302,77558762,300540197,1166803112,4537567652,17672631902,
%U 68923264412,269128937222,1052049481862,4116715363802,16123801841552
%N a(n) = binomial(2*n-5,n-2) + 2.
%C The best upper bound known for the Erdős-Szekeres problem for n >= 6.
%H G. C. Greubel, <a href="/A052473/b052473.txt">Table of n, a(n) for n = 0..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HappyEndProblem.html">Happy End Problem</a>
%F a(n) = 2 + (2^(2*n-5)*Gamma(n - 3/2))/(sqrt(Pi)*Gamma(n-1)).
%F G.f.: (x^2*(1-x) + (4 + x^2 -x^3)*sqrt(1-4*x))/(2*(1-x)*sqrt(1-4*x)). - _Eric W. Weisstein_, Jul 29 2011
%p seq( binomial(2*n-5,n-2) + 2,n=0..40); # _Robert Israel_, May 19 2019
%t Table[Binomial[2n-5, n-2] + 2, {n, 0, 30}]
%o (PARI) a(n)=binomial(2*n-5,n-2)+2 \\ _Charles R Greathouse IV_, Jul 29 2011
%o (Magma) [2 +Binomial(2*n-5,n-2): n in [0..30]]; // _G. C. Greubel_, May 18 2019
%o (Sage) [2 +binomial(2*n-5, n-2) for n in (0..30)] # _G. C. Greubel_, May 18 2019
%o (GAP) List([0..30], n-> 2+Binomial(2*n-5, n-2)) # _G. C. Greubel_, May 18 2019
%K nonn,easy
%O 0,1
%A _Eric W. Weisstein_
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