A334378 -id:A334378 - cp's OEIS frontend

A134374 a(n) = ((2n+1)!)^2.

1, 36, 14400, 25401600, 131681894400, 1593350922240000, 38775788043632640000, 1710012252724199424000000, 126513546505547170185216000000, 14797530453474819213543604224000000

Offset: 0

Artur Jasinski, Oct 22 2007

Cf. A000142, A001044, A000442, A036740, A010050, A009445, A134366, A134367, A134368, A134369, A134371, A134372, A134373, A134375, A334378.

A134374:=n->((2*n+1)!)^2; seq(A134374(n), n=0..10); # Wesley Ivan Hurt, May 02 2014

Mathematica
```
Table[((2n+1)!)^(2), {n, 0, 10}]
```

a(n) = ((2*n+1)!)^2; \\ Michel Marcus, Nov 16 2020

a(n) = A009445(n)^2 = A001044(2n+1). - Wesley Ivan Hurt, May 02 2014
From Amiram Eldar, Nov 16 2020: (Start)
Sum_{n>=0} 1/a(n) = A334378.
Sum_{n>=0} (-1)^n/a(n) = Im(BesselJ(0, 2*exp(3*Pi*i/4))). (End)

A334379 Decimal expansion of Sum_{k>=0} 1/((2*k)!)^2.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 25 2020

			1/0!^2 + 1/2!^2 + 1/4!^2 + 1/6!^2 + ... = 1.25173804073865146774451594773...

Index entries for sequences related to Bessel functions or polynomials

Cf. A010050, A049470, A070910, A073743, A091681, A197036, A334378.

RealDigits[(BesselI[0, 2] + BesselJ[0, 2])/2, 10, 110] [[1]]

suminf(k=0, 1/((2*k)!)^2) \\ Michel Marcus, Apr 26 2020

(besseli(0,2) + besselj(0,2))/2 \\ Michel Marcus, Apr 26 2020

Equals (BesselI(0,2) + BesselJ(0,2))/2.

Continued fraction: 1 + 1/(4 - 4/(145 - 144/(901 - ... - P(n-1)/((P(n) + 1) - ... )))), where P(n) = (2*n*(2*n - 1))^2. - Peter Bala, Feb 22 2024

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A134374 a(n) = ((2n+1)!)^2.

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A334379 Decimal expansion of Sum_{k>=0} 1/((2*k)!)^2.

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