A052473 a(n) = binomial(2*n-5,n-2) + 2.

2, 2, 3, 3, 5, 12, 37, 128, 464, 1718, 6437, 24312, 92380, 352718, 1352080, 5200302, 20058302, 77558762, 300540197, 1166803112, 4537567652, 17672631902, 68923264412, 269128937222, 1052049481862, 4116715363802, 16123801841552

Offset: 0

Author

Eric W. Weisstein

Keywords

• nonn
• easy

Comments

The best upper bound known for the Erdős-Szekeres problem for n >= 6.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Eric Weisstein's World of Mathematics, Happy End Problem

Programs

GAP

List([0..30], n-> 2+Binomial(2*n-5, n-2)); # G. C. Greubel, May 18 2019

Magma

[2 +Binomial(2*n-5,n-2): n in [0..30]]; // G. C. Greubel, May 18 2019

Maple

seq( binomial(2*n-5,n-2) + 2,n=0..40); # Robert Israel, May 19 2019

Mathematica

Table[Binomial[2n-5, n-2] + 2, {n, 0, 30}]

PARI

a(n)=binomial(2*n-5,n-2)+2 \\ Charles R Greathouse IV, Jul 29 2011

SageMath

[2 +binomial(2*n-5, n-2) for n in (0..30)] # G. C. Greubel, May 18 2019

Formula

a(n) = 2 + (2^(2*n-5)*Gamma(n - 3/2))/(sqrt(Pi)*Gamma(n-1)).

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