OFFSET
0,3
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.
FORMULA
a(n) = (1/2)*(C(4n, 2n) - C(2n, n)^2) = A071799(n)/2.
From Peter Luschny, Sep 05 2025: (Start)
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
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}]
(* Alternative: *)
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 *)
(* Alternative: *)
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 *)
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
David Radcliffe, Sep 04 2025
STATUS
approved