A334379 -id:A334379 - cp's OEIS frontend

A134372 a(n) = ((2n)!)^2.

1, 4, 576, 518400, 1625702400, 13168189440000, 229442532802560000, 7600054456551997440000, 437763136697395052544000000, 40990389067797283140009984000000, 5919012181389927685417441689600000000

Offset: 0

Artur Jasinski, Oct 22 2007

nonn

Cf. A000142, A001044, A000442, A036740, A010050, A134366, A134367, A134368, A134369, A134371, A134374, A134375, A334379, A334632.

Table[((2n)!)^(2), {n, 0, 10}]
((2*Range[0,20])!)^2 (* Harvey P. Dale, Jul 14 2011 *)

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

From Amiram Eldar, Nov 16 2020: (Start)
Sum_{n>=0} 1/a(n) = A334379.
Sum_{n>=0} (-1)^n/a(n) = A334632. (End)

A334378 Decimal expansion of Sum_{k>=0} 1/((2*k+1)!)^2.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 25 2020

			1/1!^2 + 1/3!^2 + 1/5!^2 + 1/7!^2 + ... = 1.027847261597415799692...
Continued fraction: 1 + 1/(36 - 36/(401 - 400/(1765 - ... - P(n-1)/((P(n) + 1) - ... )))), where P(n) = (2*n*(2*n + 1))^2 for n >= 1. - _Peter Bala_, Feb 22 2024

Index entries for sequences related to Bessel functions or polynomials

Cf. A009445, A049469, A070910, A073742, A091681, A197036, A334379.

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

suminf(k=0, 1/((2*k+1)!)^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.

A334632 Decimal expansion of Sum_{k>=0} (-1)^k / ((2*k)!)^2.

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Sep 10 2020

			0.75173418271380822855099890123074657595958657660729200273884...

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 55, page 552.

Eric Weisstein's World of Mathematics, Kelvin Functions.
Wikipedia, Kelvin functions.

Cf. A334379.

evalf(Sum((-1)^k/(2*k)!^2, k=0..infinity), 120);

RealDigits[KelvinBer[0, 2], 10, 120][[1]]
RealDigits[Re[Hypergeometric0F1Regularized[1, I]], 10, 120][[1]]
RealDigits[HypergeometricPFQ[{}, {1/2, 1/2, 1}, -1/16], 10, 120][[1]] (* Vaclav Kotesovec, Jul 19 2021 *)

PARI
```
sumalt(k=0, (-1)^k/(2*k)!^2)
```

Equals Re(BesselJ(0, 2*exp(3*Pi*i/4))).

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A134372 a(n) = ((2n)!)^2.

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A334378 Decimal expansion of Sum_{k>=0} 1/((2*k+1)!)^2.

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A334632 Decimal expansion of Sum_{k>=0} (-1)^k / ((2*k)!)^2.

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