A038534 -id:A038534 - cp's OEIS frontend

A380949 a(n) = numerator(r(n)) where r(n) = (n/2)(Pi/2)^cos(Pi(n-1))*((n/2-1/2)!/(n/2)!)^2.

0, 1, 1, 4, 9, 64, 75, 256, 1225, 16384, 19845, 65536, 160083, 1048576, 1288287, 4194304, 41409225, 1073741824, 1329696225, 4294967296, 10667118605, 68719476736, 85530896451, 274877906944, 1371086188563, 17592186044416, 21972535073125, 70368744177664, 176021737014375

Offset: 0

Peter Luschny, Feb 11 2025

			r(n) = 0, 1, 1/2, 4/3, 9/16, 64/45, 75/128, 256/175, 1225/2048, ...

Paolo Xausa, Table of n, a(n) for n = 0..1000

Cf. A380950 (denominator), A380910, A380909, A019267 (asymptotic coefficients).

Cf. A161736, A056982, A124399, A161737, A069955, A038534, A278145, A001901.

r := n -> (n/2)*(Pi/2)^cos(Pi*(n-1))*((n/2-1/2)!/(n/2)!)^2:
a := n -> numer(simplify(r(n))): seq(a(n), n = 0..28);
# Alternative:
r := n -> ifelse(n <= 1, n, (n - 1)/(n*r(n - 1))):

Join[{0}, Numerator[FoldList[(#2 - 1)/(#2*#) &, Range[30]]]] (* Paolo Xausa, Feb 14 2025 *)

Product_{k=1..n} a(k) = A380910(n) / A380909(n).
r(n) = (n - 1)/(n*r(n - 1)) for n > 1.
numerator(r(2*n)) = A161736(n).
numerator(r(2*n+1)) = A056982(n).
numerator(r(2*n+1))/4^n = A124399(n).
denominator(r(2*n-2)) = A161737(n).
denominator(r(2*n+1)) = A069955(n).
denominator(r(2*n+1))/(2*n+1) = A038534(n).
denominator(r(2*n+2))/2 = A278145(n).
denominator(r(2*n+2))/2^(2*n+1) = A001901(n).
r(n) ~ (2/Pi)^((-1)^n)*(1 - 1/(2*n) + 1/(8*n^2) + 1/(16*n^3) - 5/(128*n^4) - 23/(256*n^5) ...).

A094083 Numerators of ratio of sides of n-th triple of rectangles of unit area sum around a triangle.

Original entry on oeis.org

1, 1, 1, 4, 9, 64, 25, 256, 1225, 16384, 3969, 65536, 53361, 1048576, 184041, 4194304, 41409225, 1073741824, 147744025, 4294967296, 2133423721, 68719476736, 7775536041, 274877906944, 457028729521, 17592186044416, 1690195005625

Offset: 1

Peter J. C. Moses, Apr 30 2004

Page 13 of the link shows the type of configuration. When n is odd, the numerators 1,1,9,25,1225,3969,.. are A038534 and (A001790)^2, and the denominators 1,4,64,256,16384,65536,.. are A056982, A038533/2, and (A046161)^2. When n is even, the numerators 1,4,64,256,16384,65536,.. are A056982, A038533/2, and (A046161)^2, and the denominators 3,27,675,3675,297675,1440747,.. are 3*(A001803)^2. The limit of a(n+1)/a(n) as n(odd) tends to infinity = Pi^2/12, A072691. The limit of a(n+2)/a(n) as n tends to infinity = 1. a(n), for large odd n, tends to 2/(Pi*n). a(n), for large even n, tends to Pi/(6*n). The expansion of 2*x*EllipticK(x)/Pi gives the odd fractions. The expansion of 1/3*x*HypergeometricPFQ({1,1,1},{3/2,3/2},x) gives the even fractions.

			a(5) = a(5-2)*((5-2)/(5-1))^2 = 1/4*(3/4)^2 = 9/64

Paul Yiu, EuclideanGeometry Notes

Cf. A005386, A092936, A094084.

a[n_]:=If[OddQ[n], ((n/2-1)!)^2/(Pi*((n/2-1/2)!)^2), Pi*((n/2-1)!)^2/(12*((n/2-1/2)!)^2)] a[n_]:=If[OddQ[n], (2^(1-n)*(n-2)!!^2)/((n-1)/2)!^2, (2^(n-2)*((n-2)/2)!^2)/(3*(n-1)!!^2)] a[n_]:=((12+Pi^2+E^(I*n*Pi)*(Pi^2-12))*((n/2-1)!)^2)/(24*Pi*((n/2-1/2)!)^2) (CoefficientList[Series[(I*x*(6+Sqrt[3]*Pi)-2*x*Sqrt[3]*Log[x+Sqrt[x^2-1]])/(6*Sqrt[x^2-1]), {x, 0, 20}], x])^2

a(n)=a(n-2)*((n-2)/(n-1))^2, a(1)=1, a(2)=1/3. a(n)=((n/2-1)!)^2/(Pi*((n/2-1/2)!)^2) for n odd. a(n)=(2^(1-n)*(n-2)!!^2)/((n-1)/2)!^2 for n odd. a(n)=Pi*((n/2-1)!)^2/(12*((n/2-1/2)!)^2) for n even. a(n)=(2^(n-2)*((n-2)/2)!^2)/(3*(n-1)!!^2) for n even.

A274657 Numerators of the coefficients of z^n/n! for the expansion of hypergeometric([1/2,1/2],[1];z).

Original entry on oeis.org

1, 1, 9, 75, 3675, 59535, 2401245, 57972915, 13043905875, 418854310875, 30241281245175, 1212400457192925, 213786613951685775, 10278202593831046875, 1070401384414690453125, 60013837619516978071875, 57673297952355815927071875, 3694483615889146090857721875

Offset: 0

Wolfdieter Lang, Jul 07 2016

The denominators are given in A123854.
The main entry is A038534 (with A056982) where comments and references are given.
The complete elliptic integral of the first kind K = K(k) is (Pi/2)*hypergeometric([1/2,1/2],[1];k^2). This is also the real quarter period K of elliptic functions.

			The first rationals r(n) are: 1, 1/4, 9/32, 75/128, 3675/2048, 59535/8192, 2401245/65536, 57972915/262144, 13043905875/8388608, 418854310875/33554432, 30241281245175/268435456, ...

Cf. A038534/A056982, A123854.

With[{n = 20}, Numerator[CoefficientList[Series[2 EllipticK[x]/Pi, {x, 0, n}], x] Range[0, n]!]] (* Jan Mangaldan, Jan 04 2017 *)
Numerator[Table[Gamma[n + 1/2]^2/(Pi Gamma[n + 1]), {n, 0, 20}]] (* Li Han, Feb 05 2021 *)

a(n) = numerator(r(n)) with the rationals (in lowest terms) r(n) = (risefac(1/2,n)^2)/n! = ((2*n)!^2)/((n!^3)*2^(4*n)), with the rising factorial risefac (Pochhammer symbol).

E.g.f. for r(n) is hypergeometric([1/2,1/2],[1];z).

A275791 Triangle T(n, m) appearing in the expansion of Jacobi's elliptic function cn(u, k) divided by cos(v) in terms of the Jacobi nome q and even powers of 2cos(v) with v = u/((2/Pi)K(k)).

Original entry on oeis.org

1, -4, 1, 4, -5, 1, 0, 12, -7, 1, 4, -21, 25, -9, 1, -8, 30, -63, 42, -11, 1, 0, -44, 131, -138, 63, -13, 1, 0, 72, -246, 365, -253, 88, -15, 1, 4, -85, 425, -837, 808, -416, 117, -17, 1, -4, 85, -685, 1734, -2200, 1552, -635, 150, -19, 1, 8, -134, 1053, -3319, 5326, -4888, 2705, -918, 187, -21, 1

Offset: 0

Wolfdieter Lang, Aug 10 2016

The representation of Jacobi's elliptic cn(u, k) function in terms of quotients of theta functions of the variables q (Jacobi nome) and v = u/((2/Pi)*K(k)) with the real quarter period K is
  cn(u, k) = (theta_4(0, q)/theta_2(0, q)) * (theta_2(v, q)/theta_4(v, q)).
  This can be written either in terms of infinite sums or products. (see e.g. Tricomi, p. 176, eq. (3.87), p. 156, eq. (3.51), p. 167, eq. (3.71) with (3.71'), p. 173, eq. (3.81)).
The sums representation involves cos((2*n+1)*v) and cos(2*n*v) functions. Using Chebyshev T polynomial (A053120) one can write cn(u, k)/cos(v) = Sum_{n >= 0} q^n*Sum_{m = 0..n} T(n, m)*(2*cos(v))^(2*m).
The product representation involves directly (2*cos(v))^2 powers in the q expansion:
  cn(u, k)/cos(v) = Product_{n >= 1} ((1 - q^(2*n-1))^2 *((1 - q^(2*n))^2 + q^(2*n)*(2*cos(v))^2) / ((1 + q^(2*n))^2*((1 + q^(2*n-1))^2 - q^(2*n-1)*(2*cos(v))^2))) = Sum_{n >=0} q^n*Sum_{m = 1..n} T(n, m) * (2*cos(v))^(2*m).
For another version of this cn expansion see A274661.
For the sn(u, k)/sin(v) analog see A274662.
This entry is inspired by a paper of Bradley Klee giving an approximation to the phase space solution of the plane pendulum (see A273506).
See also the W. Lang link, equations (59) and (60).

			The triangle T(n, m) begins:
n\m 0   1    2    3    4    5    6   7   8 9
0:   1
1:  -4   1
2:   4  -5    1
3:   0  12   -7    1
4:   4 -21   25   -9     1
5:  -8  30  -63   42   -11    1
6:   0 -44  131 -138    63  -13    1
7:   0  72 -246  365  -253   88  -15   1
8:   4 -85  425 -837   808 -416  117 -17   1
9:  -4  85 -685 1734 -2200 1552 -635 150 -19 1
...
Row n=10: 8 -134 1053 -3319 5326 -4888 2705 -918 187 -21 1.
...
n=4: q^4 term of cn(u, k)/cos(v) is  4 - 21*(2*cos(v))^2 + 25*(2*cos(v))^4 - 9*(2*cos(v))^6 + (2*cos(v))^8.
One can check the identity for cn(u, k), for example for u = 1 and k = sqrt(1/2), belonging to v = 0.8472130848 and q = 0.04321391815 (Maple 10 digits), with the result from Maple's cn function cn(1, sqrt(1/2)) = 0.5959765676 (10 digits). If one takes the expansion up to q^4 inclusive one obtains 0.5959776092 (10 digits). If one goes up to q^6 inclusive one gets 0.5959765640 (10 digits).

F. Tricomi, Elliptische Funktionen (German translation by M. Krafft of: Funzioni ellittiche), Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1948.

Wolfdieter Lang, Expansions for phase space coordinates for the plane pendulum

Cf. A005797, A002103, A038534/A056982, A053120, A274661, A274662.

cn(u, k) = cos(v)*Sum_{n >= 0} q^n*Sum_{m = 0..n} T(n, m)*(2*cos(v))^(2*m), becoming an identity if q, the Jacobi nome, is replaced by exp(-Pi*K'(k)/K(k)) and v by u/((2/Pi)*K(k)) with the real and imaginary quarter periods K' and K, respectively. For the expansions of q = q(k) see A005797 or better A002103 for q = q((1-k^2)^(1/4)), and for (2/Pi)*K(k) see A038534 / A056982.

A060629 1/2+Sum_{n >= 1} a(n)x^(2n)/(4^n(2n)!) = 1/Pi*EllipticK(x).

Original entry on oeis.org

1, 27, 2250, 385875, 112521150, 49921883550, 31336679474100, 26440323306271875, 28866957423047493750, 39599692192936551926250, 66678681708870074070727500, 135213253391970365203090248750

Offset: 1

Vladeta Jovovic, Apr 14 2001

nonn

			EllipticK(x) = 1/2*Pi + 1/8*Pi*x^2 + 9/128*Pi*x^4 + 25/512*Pi*x^6 + 1225/32768*Pi*x^8 + 3969/131072*Pi*x^10 + O(x^12).

Cf. A038533, A038534.

Table[Binomial[2*n, n]*Binomial[2*n - 1, n]*(2*n)!/4^n, {n, 1, 20}] (* Vaclav Kotesovec, Nov 14 2023 *)

From Vaclav Kotesovec, Nov 14 2023: (Start)
a(n) = binomial(2*n,n) * binomial(2*n-1,n) * (2*n)! / 4^n.
a(n) ~ 2^(4*n) * n^(2*n - 1/2) / (sqrt(Pi) * exp(2*n)). (End)

A376643 Decimal expansion 4*EllipticK(4/5)/sqrt(5), where EllipticK is the complete elliptic integral of the first kind.

Original entry on oeis.org

4, 0, 3, 7, 8, 1, 1, 6, 3, 9, 9, 5, 6, 8, 4, 6, 4, 3, 1, 1, 6, 8, 0, 2, 8, 8, 7, 9, 9, 9, 7, 8, 6, 4, 9, 3, 0, 1, 3, 6, 0, 8, 3, 9, 9, 3, 4, 0, 8, 8, 0, 6, 8, 5, 7, 8, 6, 3, 4, 9, 6, 1, 5, 9, 8, 9, 7, 7, 7, 3, 8, 3, 7, 8, 6, 5, 3, 1, 9, 4, 7, 4, 4, 4, 0, 7, 7, 0, 1, 5, 0, 7, 0, 3, 3, 7, 9, 1, 9, 6, 9, 1, 0, 5, 7

Offset: 1

Amiram Eldar, Oct 01 2024

A point mass is attached to a frictionless pivot by a massless string of length L and revolves in a vertical circle about the pivot in a uniform gravitational field with an acceleration g. The slowest possible motion occurs when the tension in the string is momentarily zero at the top of the route, and the longest-possible period is then c * sqrt(L/g), where c is this constant.

			4.03781163995684643116802887999786493013608399340880...

Physics StackExchange, Maximum period of a vertically spinning ball, 2016.
Physics StackExchange, Maximal time for an object vertically swinging on a rope to complete a full circle, 2023.
Eric Weisstein's World of Mathematics, Complete Elliptic Integral of the First Kind.
Wikipedia, Elliptic integral: Complete elliptic integral of the first kind.

Constants related to similar physical problems: A019692, A038533, A038534, A175574, A256514, A309893, A310000.

RealDigits[4 * EllipticK[4/5] / Sqrt[5], 10, 120][[1]]

PARI
```
4*ellK(sqrt(4/5))/sqrt(5)
```

Equals 2 * Integral_{0..Pi} (1/sqrt(3 + 2*cos(x))) dx.

cp's OEIS Frontend

A380949 a(n) = numerator(r(n)) where r(n) = (n/2)*(Pi/2)^cos(Pi*(n-1))*((n/2-1/2)!/(n/2)!)^2.

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A094083 Numerators of ratio of sides of n-th triple of rectangles of unit area sum around a triangle.

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A274657 Numerators of the coefficients of z^n/n! for the expansion of hypergeometric([1/2,1/2],[1];z).

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A275791 Triangle T(n, m) appearing in the expansion of Jacobi's elliptic function cn(u, k) divided by cos(v) in terms of the Jacobi nome q and even powers of 2*cos(v) with v = u/((2/Pi)*K(k)).

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A060629 1/2+Sum_{n >= 1} a(n)*x^(2*n)/(4^n*(2*n)!) = 1/Pi*EllipticK(x).

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A376643 Decimal expansion 4*EllipticK(4/5)/sqrt(5), where EllipticK is the complete elliptic integral of the first kind.

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A380949 a(n) = numerator(r(n)) where r(n) = (n/2)(Pi/2)^cos(Pi(n-1))*((n/2-1/2)!/(n/2)!)^2.

A275791 Triangle T(n, m) appearing in the expansion of Jacobi's elliptic function cn(u, k) divided by cos(v) in terms of the Jacobi nome q and even powers of 2cos(v) with v = u/((2/Pi)K(k)).

A060629 1/2+Sum_{n >= 1} a(n)x^(2n)/(4^n(2n)!) = 1/Pi*EllipticK(x).