A143623 -id:A143623 - cp's OEIS frontend

A226731 a(n) = (2n - 1)!/(2n).

20, 630, 36288, 3326400, 444787200, 81729648000, 19760412672000, 6082255020441600, 2322315553259520000, 1077167364120207360000, 596585001666576384000000, 388888194657798291456000000

Offset: 3

Wesley Ivan Hurt, Jun 15 2013

For n < 3, the formula does not produce an integer.

The ratio of the product of the partition parts of 2n into exactly two parts to the sum of the partition parts of 2n into exactly two parts. For example, a(3) = 20, and 2*3 = 6 has 3 partitions into exactly two parts: (5,1), (4,2), (3,3). Forming the ratio of product to sum (of parts), we have (5*1*4*2*3*3)/(5+1+4+2+3+3) = 360/18 = 20. - Wesley Ivan Hurt, Jun 24 2013

			a(3) = (2*3 - 1)!/(2*3) = 5!/6 = 120/6 = 20.

Seiichi Manyama, Table of n, a(n) for n = 3..225
Index entries for sequences related to partitions.

Cf. A001105, A002674, A005843, A009445, A093353, A143623, A211374.

seq((2*k-1)!/(2*k),k=1..20); # Wesley Ivan Hurt, Jun 24 2013

Table[(2n-1)!/(2n),{n,3,20}] (* Harvey P. Dale, Jun 19 2013 *)

a(n) = (2*n-1)!/(2*n); \\ Michel Marcus, Nov 01 2016

a(n) = A009445(n-1)/A005843(n) = A002674(n)/A001105(n). - Wesley Ivan Hurt, Jun 24 2013
a(n) ~ sqrt(Pi)*2^(2*n-1)*n^(2*n-3/2)/exp(2*n). - Ilya Gutkovskiy, Nov 01 2016
From Amiram Eldar, Mar 10 2022: (Start)
Sum_{n>=3} 1/a(n) = e - 8/3.
Sum_{n>=3} (-1)^(n+1)/a(n) = cos(1) + sin(1) - 4/3. (End)

A165233 Signed denominators of terms in series expansion of cos(x)+sin(x).

Original entry on oeis.org

1, 1, -2, -6, 24, 120, -720, -5040, 40320, 362880, -3628800, -39916800, 479001600, 6227020800, -87178291200, -1307674368000, 20922789888000, 355687428096000, -6402373705728000, -121645100408832000, 2432902008176640000

Offset: 0

Jaume Oliver Lafont, Sep 09 2009

Sum(n>=0,1/a(n))=cos(1)+sin(1).
Sum(n>=0,(Pi/4)^n/a(n))=sqrt(2).
Numerators are in A000012. - Alois P. Heinz, Jan 20 2016

Cf. A000012, A000142, A133942, A057077, A143623, A002193.

Sign@ # Denominator@ # & /@ CoefficientList[Series[Cos@ x + Sin@ x, {x, 0, 20}], x] (* Michael De Vlieger, Oct 08 2016 *)

PARI
```
a(n)=(-1)^(n\2)*n!
```

a(n)=(-1)^floor(n/2)*n! = A057077(n)*A000142(n).
a(n)=(sin(n*Pi/2)+cos(n*Pi/2))*n!.
a(n)=sqrt(2)*sin((2n+1)*Pi/4)*n!.
a(n)=sqrt(2)*cos((2n-1)*Pi/4)*n!.
G.f. Q(0) where Q(k)= 1 + x*(4*k+1)/(1 + 2*x*(2*k+1)/(1 - 2*x*(2*k+1) - x*(4*k+3)/(1 + x*(4*k+3) - 4*x*(k+1)/(4*x*(k+1) - 1/Q(k+1))))); (continued fraction, 3rd kind, 5-step). - Sergei N. Gladkovskii, Aug 15 2012
E.g.f.: (1 + x)/(1 + x^2). - Ilya Gutkovskiy, Oct 08 2016

A371934 Decimal expansion of Sum_{k>=0} (-1)^k / ((k+1)(2k)!).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Apr 24 2024

			0.7635465813520724481068778581465512067097...

Cf. A143623, A334397.

hypergeom([1], [1/2, 2], -1/4) ; evalf(%) ; # R. J. Mathar, Jul 03 2024

s = N[Sum[(-1)^k/((k + 1) (2 k)!), {k, 0, Infinity}], 120]
First[RealDigits[s]]

Equals 2*(A143623 - 1).

cp's OEIS Frontend

A226731 a(n) = (2n - 1)!/(2n).

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A165233 Signed denominators of terms in series expansion of cos(x)+sin(x).

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A371934 Decimal expansion of Sum_{k>=0} (-1)^k / ((k+1)(2k)!).

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cp's OEIS Frontend

A226731 a(n) = (2n - 1)!/(2n).

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Maple

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A165233 Signed denominators of terms in series expansion of cos(x)+sin(x).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Formula

A371934 Decimal expansion of Sum_{k>=0} (-1)^k / ((k+1)*(2*k)!).

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Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

A371934 Decimal expansion of Sum_{k>=0} (-1)^k / ((k+1)(2k)!).