A009120 -id:A009120 - cp's OEIS frontend

A166338 a(n) = (4*n)!/n!.

1, 24, 20160, 79833600, 871782912000, 20274183401472000, 861733891296165888000, 60493719168990845337600000, 6526062423950732395020288000000, 1025113885554181044609786839040000000, 224844379201911853600532206127677440000000, 66595307609539060446820030939720014888960000000

Offset: 0

Karol A. Penson, Oct 12 2009

nonn

Integral representation as n-th moment of a positive function on a positive halfaxis (solution of the Stieltjes moment problem), in Maple notation: a(n) = Integral_{x=0..oo} ( x^n*((1/16*(2*Pi^(3/2)*sqrt(2)*hypergeom([], [1/2, 3/4], -(1/256)*x)*sqrt(x) -2*Pi*sqrt(2)*hypergeom([], [3/4, 5/4], -(1/256)*x)*Gamma(3/4)*x^(3/4) +sqrt(Pi)*Gamma(3/4)^2*hypergeom([], [5/4, 3/2], -(1/256)*x)*x))*sqrt(2)/(Gamma(3/4)*x^(5/4)*Pi^(3/2))) ).

This solution may not be unique.

G. C. Greubel, Table of n, a(n) for n = 0..100

Cf. A009120, A061924.

[Factorial(4*n) / Factorial(n): n in [0..15]]; // Vincenzo Librandi, May 10 2016

A166338_list := proc(n) u:=z^(1/4);(cosh(u)+cos(u))/2:series(%,z,n+2):
seq(1/(i!*coeff(%,z,i)),i=0..n) end: A166338_list(9); # Peter Luschny, Jul 12 2012

Table[(4n)!/n!,{n,0,10}] (* Harvey P. Dale, May 30 2015 *)

G.f.: Sum_{n>=0} a(n)*x^n/(n!)^3 = hypergeom([1/4, 1/2, 3/4], [1, 1], 256*x).
a(n) ~ 2*(1-1/(16*n)+1/(512*n^2)+331/(122880*n^3)+O(1/n^4)))*(2^n)^8/(((1/n)^n)^3*(exp(n))^3).
1/a(n) = n!*[x^n](cosh(x^(1/4))+cos(x^(1/4)))/2. - Peter Luschny, Jul 12 2012
From Seiichi Manyama, May 25 2025: (Start)
a(n) = RisingFactorial(n+1,3*n).
a(n) = (3*n)! * [x^(3*n)] 1/(1 - x)^(n+1). (End)

A101485 a(n) = (4n)! / ( 4^n * (2n)! ).

Original entry on oeis.org

1, 3, 105, 10395, 2027025, 654729075, 316234143225, 213458046676875, 191898783962510625, 221643095476699771875, 319830986772877770815625, 563862029680583509947946875, 1192568192774434123539907640625, 2980227913743310874726229193921875

Offset: 0

Ralf Stephan, Jan 21 2005

Bisection of A001147. Odd part of A009120.

seq(4^n*pochhammer(1/2,2*n), n=0..12); # Peter Luschny, Jul 05 2011

f[n_] := 4^n*Pochhammer[1/2, 2 n]; Array[f, 13, 0] (* Robert G. Wilson v, Jul 05 2011 *)

a(n)=(4*n)!/(2*n)!>>(2*n) \\ Charles R Greathouse IV, Jul 06 2011

sin(arcsin(2x)/2) = x + 3x^3/3! + 105x^5/5! + 10395x^7/7! + ...
E.g.f.: cosh(x^2/2). - Paul Barry, Sep 28 2010
a(n) = 4^n*Gamma(2*n+1/2) / Gamma(1/2). - Peter Luschny, Jul 05 2011
Hypergeom. recurrence: a(n) -(4*n-1)*(4*n-3)*a(n-1)=0. - R. J. Mathar, Sep 21 2012
Sum_{n>=0} 1/a(n) = 1 + (1/2) * sqrt(e*Pi/2) * erf(1/sqrt(2)) - (1/2) * sqrt(Pi/(2*e)) * erfi(1/sqrt(2)), where erf is the error function and erfi is the imaginary error function. - Amiram Eldar, Jan 08 2023

A349468 a(n) = (4n)! / (n! (2*n)!).

Original entry on oeis.org

1, 12, 840, 110880, 21621600, 5587021440, 1799020903680, 693908062848000, 311911674250176000, 160114659448423680000, 92418181433630148096000, 59248455951814527670272000, 41770161446029242007541760000, 32118041062654484854414417920000, 26749739913610806671605150924800000

Offset: 0

Ilya Gutkovskiy, Nov 18 2021

Cf. A000897, A000984, A001448, A001813, A009120, A166338.

Table[(4 n)!/(n! (2 n)!), {n, 0, 14}]
nmax = 14; CoefficientList[Series[2 EllipticK[16 Sqrt[x]/(1 + 8 Sqrt[x])]/(Pi Sqrt[1 + 8 Sqrt[x]]), {x, 0, nmax}], x] Range[0, nmax]!
Table[SeriesCoefficient[D[1/Sqrt[1 - 4 x], {x, n}], {x, 0, n}], {n, 0, 14}]

a(n) = (4*n)! / (n! * (2*n)!) \\ Andrew Howroyd, Nov 20 2021

E.g.f.: 2 * EllipticK( 16*sqrt(x) / (1 + 8*sqrt(x)) ) / (Pi * sqrt(1 + 8*sqrt(x))).
a(n) is the coefficient of x^n in expansion of d^n/dx^n g(x), where g(x) is the g.f. of central binomial coefficients (A000984).
a(n) = n! * A000897(n) = A009120(n) / n! = A166338(n) / (2*n)! = A001448(n) * A001813(n).
a(n) ~ 64^n * n^(n-1/2) / (sqrt(Pi) * exp(n)).
D-finite with recurrence n*a(n) -4*(4*n-1)*(4*n-3)*a(n-1)=0. - R. J. Mathar, Mar 06 2022

cp's OEIS Frontend

A166338 a(n) = (4*n)!/n!.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

A101485 a(n) = (4n)! / ( 4^n * (2n)! ).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A349468 a(n) = (4n)! / (n! (2*n)!).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

cp's OEIS Frontend

A166338 a(n) = (4*n)!/n!.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

A101485 a(n) = (4n)! / ( 4^n * (2n)! ).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A349468 a(n) = (4*n)! / (n! * (2*n)!).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A349468 a(n) = (4n)! / (n! (2*n)!).