A218708 -id:A218708 - cp's OEIS frontend

A366107 a(n) = Sum_{i=0..floor(q(n)/3)} binomial(n-3(i+1), q(n)-3i) with q(n) = ceiling((n-3)/2).

1, 1, 2, 3, 6, 11, 21, 39, 75, 141, 273, 519, 1009, 1933, 3770, 7263, 14202, 27479, 53846, 104543, 205216, 399543, 785460, 1532779, 3017106, 5899167, 11624580, 22766607, 44905518, 88073091, 173863965, 341425551, 674506059, 1326019653, 2621371005, 5158412943, 10203609597

Offset: 3

Stefano Spezia, Sep 29 2023

nonn

Gábor Czédli, Minimum-sized generating sets of the direct powers of the free distributive lattice on three generators and a Sperner theorem, arXiv:2309.13783 [math.CO], 2023. See formulas (3.5) at p. 4 and (4.15) at p. 8.

Cf. A004526, A218708, A366108, A366109.

q[n_]:=Ceiling[(n-3)/2]; a[n_]:=Sum[Binomial[n-3(i+1),q[n]-3i], {i,0,Floor[q[n]/3]}]; Array[a,37,3]

a(n) = my(q=ceil((n-3)/2)); sum(i=0, q\3, binomial(n-3*(i+1), q-3*i)); \\ Michel Marcus, Sep 30 2023

From Remark 3.4 at p. 5 in Czédli: (Start)
A366108(n)/a(n) ~ 7/4.
A366109(n)/a(n) ~ 7/6. (End)
a(n) ~ c*2^(n+1)/sqrt(n), with c = 1/(7*sqrt(2*Pi)) = (2/7)* A218708.

A366108 a(n) = floor(binomial(n-1, floor((n-1)/2))/2).

Original entry on oeis.org

1, 1, 3, 5, 10, 17, 35, 63, 126, 231, 462, 858, 1716, 3217, 6435, 12155, 24310, 46189, 92378, 176358, 352716, 676039, 1352078, 2600150, 5200300, 10029150, 20058300, 38779380, 77558760, 150270097, 300540195, 583401555, 1166803110, 2268783825, 4537567650, 8836315950

Offset: 3

Stefano Spezia, Sep 29 2023

nonn

Gábor Czédli, Minimum-sized generating sets of the direct powers of the free distributive lattice on three generators and a Sperner theorem, arXiv:2309.13783 [math.CO], 2023. See formulas (3.6) at p. 4 and (4.15) at p. 8.

Cf. A001405, A004526, A218708, A335322, A366107, A366109.

a[n_]:=Floor[Binomial[n-1,Floor[(n-1)/2]]/2]; Array[a,36,3]

a(n) = binomial(n-1, (n-1)\2)\2; \\ Michel Marcus, Sep 30 2023

a(n)/A366107(n) ~ 7/4 (see Remark 3.4 at p. 5 in Czédli).

a(n) ~ c*2^n/sqrt(n), with c = 1/(2*sqrt(2*Pi)) = A218708.

A366109 a(n) = floor(n!(3floor(n/2)!ceiling(n/2)! + 3floor((n+2)/2)!ceiling((n-2)/2)! - 6floor(n/2)!*ceiling((n-2)/2)!)^(-1)).

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 26, 46, 92, 168, 333, 616, 1225, 2288, 4558, 8580, 17107, 32413, 64664, 123170, 245832, 470288, 938943, 1802770, 3600207, 6933733, 13849778, 26744400, 53429368, 103411680, 206621384, 400720260, 800747232, 1555737480, 3109074130, 6050090200, 12091800773

Offset: 3

Stefano Spezia, Sep 29 2023

nonn

Gábor Czédli, Minimum-sized generating sets of the direct powers of the free distributive lattice on three generators and a Sperner theorem, arXiv:2309.13783 [math.CO], 2023. See formulas (3.6) at p. 4 and (4.15) at p. 8.

Cf. A000142, A004526, A081123, A218708, A366107, A366108.

a[n_]:=Floor[n!(3Floor[n/2]!Ceiling[n/2]! + 3Floor[(n+2)/2]!Ceiling[(n-2)/2]! - 6Floor[n/2]!Ceiling[(n-2)/2]!)^(-1)]; Array[a,37,3]

a(n)/A366107(n) ~ 7/6 (see Remark 3.4 at p. 5 in Czédli).

a(n) ~ c*2^n/sqrt(n), with c = 1/(3*sqrt(2*Pi)) = (2/3)*A218708.

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

Original entry on oeis.org

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

Offset: 0

David Radcliffe, Sep 04 2025

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.

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.

Cf. A071799, A036910, A002894, A218708.

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

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 *)

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

cp's OEIS Frontend

A366107 a(n) = Sum_{i=0..floor(q(n)/3)} binomial(n-3*(i+1), q(n)-3*i) with q(n) = ceiling((n-3)/2).

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A366108 a(n) = floor(binomial(n-1, floor((n-1)/2))/2).

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A366109 a(n) = floor(n!*(3*floor(n/2)!*ceiling(n/2)! + 3*floor((n+2)/2)!*ceiling((n-2)/2)! - 6*floor(n/2)!*ceiling((n-2)/2)!)^(-1)).

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Mathematica

Formula

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

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Mathematica

Formula

A366107 a(n) = Sum_{i=0..floor(q(n)/3)} binomial(n-3(i+1), q(n)-3i) with q(n) = ceiling((n-3)/2).

A366109 a(n) = floor(n!(3floor(n/2)!ceiling(n/2)! + 3floor((n+2)/2)!ceiling((n-2)/2)! - 6floor(n/2)!*ceiling((n-2)/2)!)^(-1)).