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

0, 1, 17, 262, 3985, 60626, 925190, 14168988, 217721745, 3355615450, 51855874642, 803232328548, 12467572005382, 193873026294052, 3019674502600220, 47101568276955512, 735663252850019217, 11503661742608944170, 180077229781765344602, 2821666487800835457300

Offset: 0

Author

David Radcliffe, Sep 04 2025

Keywords

• nonn
• easy
• new

Comments

a(n) is the number of subsets of {1,...,4n} of size 2n containing at least n+1 elements from {1,...,2n}.

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.

Links

• Rudolf Ahlswede and Levon H. Khachatrian, The complete intersection theorem for systems of finite sets. European J. Combin. 18 (1997), 125-136.
• Thomas Bloom, Problem 83, Erdős Problems.
• P. Erdős, Chao Ko, and R. Rado, Intersection theorems for systems of finite sets. Quart. J. Math. Oxford Ser. (2) (1961), 313-320.

Crossrefs

Cf. A071799, A036910, A002894, A218708.

Programs

Maple

seq(add(binomial(2*n, k)^2, k=0..(n-1)), n=0..20);
# or
gf := (1/2)*((sqrt(1 + sqrt(1 - 16*x)))/(sqrt(2 - 32*x)) - hypergeom([1/2, 1/2], [1], 16*x)):
ser := series(gf, x, 20): seq(coeff(ser, x, n), n = 0..19);  # Peter Luschny, Sep 05 2025

Mathematica

Table[(Binomial[4n, 2n] - Binomial[2n, n]^2)/2, {n, 0, 20}]
(* or *)
gf[x_] := (Sqrt[1 + Sqrt[1 - 16 x]])/(2 Sqrt[2 - 32 x] ) - EllipticK[16 x]/Pi;
CoefficientList[Series[gf[x], {x, 0, 19}], x]  (* Peter Luschny, Sep 05 2025 *)
(* or *)
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 *)

Formula

a(n) = (1/2)*(C(4n, 2n) - C(2n, n)^2) = A071799(n)/2.

From Peter Luschny, Sep 05 2025: (Start)

a(n) = A036910(n) - A002894(n).

a(n) = [x^n]((1/2)*((sqrt(1 + sqrt(1 - 16*x)))/(sqrt(2 - 32*x)) - hypergeom([1/2, 1/2], [1], 16*x))).

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.

a(n) ~ 16^n/sqrt(8*Pi*n) = A218708*16^n/sqrt(n). (End)

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