cp's OEIS Frontend

A387635 a(n) = Sum_{k=0..n-1} binomial(2*n, k)^2.

%I A387635 #27 Sep 06 2025 05:29:27
%S A387635 0,1,17,262,3985,60626,925190,14168988,217721745,3355615450,
%T A387635 51855874642,803232328548,12467572005382,193873026294052,
%U A387635 3019674502600220,47101568276955512,735663252850019217,11503661742608944170,180077229781765344602,2821666487800835457300
%N A387635 a(n) = Sum_{k=0..n-1} binomial(2*n, k)^2.
%C A387635 a(n) is the number of subsets of {1,...,4n} of size 2n containing at least n+1 elements from {1,...,2n}.
%C A387635 Also the maximum size of a family of 2n-subsets of a 4n-set such that every pairwise intersection has at least two elements. This was conjectured by Erdős, Ko, and Rado, and proved by Ahlswede and Khachatrian.
%H A387635 Rudolf Ahlswede and Levon H. Khachatrian, <a href="https://www.math.uni-bielefeld.de/ahlswede/homepage/public/122.pdf">The complete intersection theorem for systems of finite sets</a>. European J. Combin. 18 (1997), 125-136.
%H A387635 Thomas Bloom, <a href="https://www.erdosproblems.com/83">Problem 83</a>, Erdős Problems.
%H A387635 P. Erdős, Chao Ko, and R. Rado, <a href="https://combinatorica.hu/~p_erdos/1961-07.pdf">Intersection theorems for systems of finite sets</a>. Quart. J. Math. Oxford Ser. (2) (1961), 313-320.
%F A387635 a(n) = (1/2)*(C(4n, 2n) - C(2n, n)^2) = A071799(n)/2.
%F A387635 From _Peter Luschny_, Sep 05 2025: (Start)
%F A387635 a(n) = A036910(n) - A002894(n).
%F A387635 a(n) = [x^n]((1/2)*((sqrt(1 + sqrt(1 - 16*x)))/(sqrt(2 - 32*x)) - hypergeom([1/2, 1/2], [1], 16*x))).
%F A387635 a(n) = [x^n]((sqrt(1 + sqrt(1 - 16*x)))/(2*sqrt(2 - 32*x)) - EllipticK((4*sqrt(x))^m)/Pi) where m = 1 if the Maple conventions and m = 2 if the Mathematica conventions are followed.
%F A387635 a(n) ~ 16^n/sqrt(8*Pi*n) = A218708*16^n/sqrt(n). (End)
%F A387635 a(n) = [x^n] sqrt(1+sqrt(1-16*x))/(2*sqrt(2-32*x)) - 1/(2*AGM(1,sqrt(1-16*x))). - _Vaclav Kotesovec_, Sep 06 2025
%p A387635 seq(add(binomial(2*n, k)^2, k=0..(n-1)), n=0..20);
%p A387635 # or
%p A387635 gf := (1/2)*((sqrt(1 + sqrt(1 - 16*x)))/(sqrt(2 - 32*x)) - hypergeom([1/2, 1/2], [1], 16*x)):
%p A387635 ser := series(gf, x, 20): seq(coeff(ser, x, n), n = 0..19);  # _Peter Luschny_, Sep 05 2025
%t A387635 Table[(Binomial[4n, 2n] - Binomial[2n, n]^2)/2, {n, 0, 20}]
%t A387635 (* or *)
%t A387635 gf[x_] := (Sqrt[1 + Sqrt[1 - 16 x]])/(2 Sqrt[2 - 32 x] ) - EllipticK[16 x]/Pi;
%t A387635 CoefficientList[Series[gf[x], {x, 0, 19}], x]  (* _Peter Luschny_, Sep 05 2025 *)
%t A387635 (* or *)
%t A387635 CoefficientList[Series[(Sqrt[1 + Sqrt[1 - 16*x]])/(2*Sqrt[2 - 32*x]) - 1/(2*ArithmeticGeometricMean[1, Sqrt[1 - 16*x]]), {x, 0, 19}], x] (* _Vaclav Kotesovec_, Sep 06 2025 *)
%Y A387635 Cf. A071799, A036910, A002894, A218708.
%K A387635 nonn,easy,new
%O A387635 0,3
%A A387635 _David Radcliffe_, Sep 04 2025