A248759 -id:A248759 - cp's OEIS frontend

A064350 a(n) = (3*n)!/n!.

1, 6, 360, 60480, 19958400, 10897286400, 8892185702400, 10137091700736000, 15388105201717248000, 30006805143348633600000, 73096577329197271449600000, 217535414131691079834009600000, 776601428450137155007414272000000, 3275704825202678519821273399296000000

Offset: 0

Karol A. Penson, Sep 18 2001

Also a(n) = (((n)!)^2)*A006480(n). [corrected by Johannes W. Meijer, Mar 02 2009]

a(n) is the number of ways to partition the set {1,2,...,3n} into n blocks of size 3 and then linearly order the elements within each block. - Geoffrey Critzer, Dec 30 2012

Harry J. Smith, Table of n, a(n) for n = 0..70
Karol A. Penson and Allan I. Solomon, Coherent states from combinatorial sequences, in: E. Kapuscik and A. Horzela (eds.), Quantum theory and symmetries, World Scientific, 2002, pp. 527-530; arXiv preprint, arXiv:quant-ph/0111151, 2001.

Cf. A006480, A248759.
From Johannes W. Meijer, Mar 07 2009: (Start)
Equals A001525*3!
Equals row sums of A157704 and A157705. (End)

Table[(3n)!/n!,{n,0,20}]  (* Geoffrey Critzer, Dec 30 2012 *)

{ t=f=1; for (n=0, 70, if (n, t*=3*n*(3*n - 1)*(3*n - 2); f*=n); write("b064350.txt", n, " ", t/f) ) } \\ Harry J. Smith, Sep 12 2009

Integral representation as n-th moment of a positive function on a positive half-axis: a(n) = Integral_{x=0..oo} x^n*BesselK(1/3, 2*sqrt(x/27))/(3*Pi*sqrt(x)) dx, n >= 0.
A recursive formula: a(n) = (27 * (n - 1)^2 + 27 * (n - 1) + 6) * a(n - 1) with a(0) = 1. An explicit formula following from the recursion equation: a(n) = (3/2)*27^n*GAMMA(n+2/3)*GAMMA(n+1/3)/(Pi*3^(1/2)). - Thomas Wieder, Nov 15 2004
E.g.f.: (of aerated sequence) 2*cos(arcsin((3*sqrt(3)*x/2)/3))/sqrt(4-27*x^2). - Paul Barry, Jul 27 2010
E.g.f.: (with interpolated zeros) exp(x^3). - Geoffrey Critzer, Dec 30 2012
Sum_{n>=1} 1/a(n) = A248759. - Amiram Eldar, Nov 15 2020

a(11) from Harry J. Smith, Sep 12 2009

A248760 Decimal expansion of sum_{n >= 1} (2n)!/(3n)!.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Oct 13 2014

			0.368737820294649904089777296165740342098371978814698491068782574589059...

Cf. A214869, A248696, A248759, A248761.

evalf(sum((2n)!/(3n)!, n=1..infinity), 120); # Vaclav Kotesovec, Oct 19 2014

u = N[Sum[(2 n)!/(3 n)!, {n, 1, 300}], 120]
RealDigits[u]  (* A248760 *)
N[HypergeometricPFQ[{1, 3/2}, {4/3, 5/3}, 4/27]/3, 120] (* Vaclav Kotesovec, Nov 15 2020 *)

suminf(n=1, (2*n)!/(3*n)!) \\ Michel Marcus, Oct 19 2014

A248761 Decimal expansion of sum_{n >= 1} 1/sqrt(n!).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Oct 13 2014

			2.46950631452104756247563674466015025768975618399496919967792570...

Cf. A214869, A248696, A248759, A248760.

evalf(sum(1/sqrt(n!), n=1..infinity), 120); # Vaclav Kotesovec, Oct 19 2014

u = N[Sum[1/Sqrt[n!], {n, 1, 200}], 100]
RealDigits[u]  (* A248761 *)

suminf(n=1, (1/sqrt(n!))) \\ Michel Marcus, Oct 18 2014

cp's OEIS Frontend

A064350 a(n) = (3*n)!/n!.

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A248760 Decimal expansion of sum_{n >= 1} (2n)!/(3n)!.

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A248761 Decimal expansion of sum_{n >= 1} 1/sqrt(n!).

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