A068447 -id:A068447 - cp's OEIS frontend

A052277 a(n) = (4n+2)!/2^(2n+1).

1, 90, 113400, 681080400, 12504636144000, 548828480360160000, 49229914688306352000000, 8094874872198213459360000000, 2252447502438386084347676160000000, 997586474354936812896742294502400000000, 669959124447288464805194190141921792000000000

Offset: 0

N. J. A. Sloane, Feb 05 2000

J. M. Borwein, D. M. Bradley, and D. J. Broadhurst, Evaluations of k-fold Euler/Zagier sums: a compendium of results for arbitrary k, arXiv:hep-th/9611004, 1996.
Roudy El Haddad, Multiple Sums and Partition Identities, arXiv:2102.00821 [math.CO], 2021.
Roudy El Haddad, A generalization of multiple zeta value. Part 2: Multiple sums. Notes on Number Theory and Discrete Mathematics, 28(2), 2022, 200-233, DOI: 10.7546/nntdm.2022.28.2.200-233.

Cf. A002432 (denominators of zeta(2*n)/Pi^(2*n)).

Cf. A068447, A067912, A013662 (zeta(4)).

Table[(4n+2)!/2^(2n+1), {n, 0, 10}] (* Amiram Eldar, Feb 25 2022 *)

a(n) = (4*n+2)!/2^(2*n+1); \\ Michel Marcus, Feb 20 2022

sin(x)*sinh(x) = Sum_{n>=0} (-1)^n*x^(4n+2)/a(n). - Benoit Cloitre, Feb 02 2002
a(n) = Pi^(4n)/Zeta({4}n) where ({4}_n) is the standard multiple zeta values notation for (4, ..., 4) where the multiplicity of 4 is n. - _Roudy El Haddad, Feb 19 2022
From Amiram Eldar, Feb 25 2022: (Start)
Sum_{n>=0} 1/a(n) = (cosh(sqrt(2)) - cos(sqrt(2)))/2.
Sum_{n>=0} (-1)^n/a(n) = sin(1)*sinh(1). (End)

A333972 Decimal expansion of Pi^6/540 = zeta(2) * zeta(4).

Original entry on oeis.org

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

Offset: 1

Bernard Schott, Sep 29 2020

Compare 1st formula with Sum_{m>0, q>0} 1/(m^2*q^2) = Pi^4/36 = (zeta(2))^2 = A098198.

			1.78035035847278599450040637713411092382818060755749373322421516...

Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année MP, Dunod, 1997, Exercice 3.22, p. 275.

Index entries for transcendental numbers

Cf. A001157, A013661, A046951, A068447, A098198.

Maple
```
evalf(Pi^6/540,120);
```

RealDigits[Pi^6/540, 10, 100][[1]] (* Amiram Eldar, Sep 29 2020 *)

PARI
```
Pi^6/540 \\ Michel Marcus, Sep 30 2020
```

Equals Sum_{m>0, q>0, m | q} 1/(m^2*q^2).
Equals A013661 * A068447.
Equals Sum_{k>=1} sigma_2(k)/k^4. - Amiram Eldar, Sep 30 2020
Equals Sum_{k>=1} A046951(k)/k^2. - Amiram Eldar, Jan 25 2024

A067277 Factorial expansion of zeta(3): zeta(3) = Sum_{n>=1} a(n)/n!.

Original entry on oeis.org

1, 0, 1, 0, 4, 1, 3, 2, 8, 4, 0, 11, 11, 10, 9, 4, 2, 11, 5, 12, 16, 12, 6, 3, 22, 22, 12, 14, 23, 1, 24, 24, 12, 14, 1, 27, 14, 26, 21, 16, 22, 14, 6, 19, 12, 12, 36, 22, 32, 38, 10, 1, 14, 51, 9, 6, 51, 26, 50, 25, 30, 44, 19, 49, 12, 17, 24, 55, 17, 47, 11, 8, 43, 71, 43, 16, 76

Offset: 1

Benoit Cloitre, Mar 10 2002

			zeta(3) = 1 + 1/3! + 4/5! + 1/6! + 3/7! + 2/8! + 8/9! + 4/10! + ...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A067279 (zeta(2)), A068447 (zeta(4)), A068454 (zeta(5)), A068455 (zeta(6)), A068456 (zeta(7)), A068457 (zeta(8)), A068458 (zeta(9)), A068459 (zeta(10)).

SetDefaultRealField(RealField(250)); L:=RiemannZeta(); [Floor(Evaluate(L,3))] cat [Floor(Factorial(n)*Evaluate(L,3)) - n*Floor(Factorial((n-1))*Evaluate(L,3)) : n in [2..80]]; // G. C. Greubel, Nov 26 2018

With[{b = Zeta[3]}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]] (* G. C. Greubel, Nov 26 2018 *)

default(realprecision, 250); b = zeta(3); for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", ")) \\ G. C. Greubel, Nov 26 2018

def A067279(n):
    if (n==1): return floor(zeta(3))
    else: return expand(floor(factorial(n)*zeta(3)) - n*floor(factorial(n-1)*zeta(3)))
[A067279(n) for n in (1..80)] # G. C. Greubel, Nov 26 2018

a(n) = floor(n!*zeta(3)) - n*floor((n-1)!*zeta(3)), with a(1)=1, for n > 1.

A067279 Factorial expansion of zeta(2) : zeta(2) = Sum_{n>=1} a(n)/n!.

Original entry on oeis.org

1, 1, 0, 3, 2, 2, 2, 3, 6, 6, 8, 1, 11, 12, 7, 6, 13, 7, 3, 2, 2, 2, 9, 20, 9, 16, 11, 0, 12, 13, 19, 25, 26, 31, 18, 24, 21, 32, 12, 34, 22, 24, 13, 14, 41, 20, 34, 29, 22, 40, 50, 4, 33, 50, 39, 8, 15, 24, 14, 59, 40, 3, 9, 29, 27, 14, 18, 39, 59, 44, 28, 30, 35, 5, 64, 20, 18

Offset: 1

Benoit Cloitre, Mar 10 2002

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A067277 (zeta(3)), A068447 (zeta(4)), A068454 (zeta(5)), A068455 (zeta(6)), A068456 (zeta(7)), A068457 (zeta(8)), A068458 (zeta(9)), A068459 (zeta(10)).

SetDefaultRealField(RealField(250)); L:=RiemannZeta(); [Floor(Evaluate(L,2))] cat [Floor(Factorial(n)*Evaluate(L,2)) - n*Floor(Factorial((n-1))*Evaluate(L,2)) : n in [2..80]]; // G. C. Greubel, Nov 26 2018

With[{b = Zeta[2]}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]] (* G. C. Greubel, Nov 26 2018 *)

default(realprecision, 250); b = zeta(2); for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", ")) \\ G. C. Greubel, Nov 26 2018

def A067279(n):
    if (n==1): return floor(zeta(2))
    else: return expand(floor(factorial(n)*zeta(2)) - n*floor(factorial(n-1)*zeta(2)))
[A067279(n) for n in (1..80)] # G. C. Greubel, Nov 26 2018

a(n) = floor(n!*zeta(2)) - n*floor((n-1)!*zeta(2)), for n>=2.

a(1) corrected by G. C. Greubel, Nov 26 2018

A068454 Factorial expansion of zeta(5) = Sum_{n>=1} a(n)/n!, with a(n) as large as possible.

Original entry on oeis.org

1, 0, 0, 0, 4, 2, 4, 0, 8, 3, 4, 9, 10, 5, 3, 12, 4, 1, 10, 0, 6, 19, 0, 19, 10, 21, 19, 16, 3, 27, 24, 12, 12, 14, 7, 33, 27, 15, 28, 15, 7, 15, 7, 21, 13, 29, 16, 44, 39, 27, 39, 17, 6, 18, 2, 21, 21, 35, 29, 12, 13, 6, 39, 14, 1, 23, 55, 34, 10, 42, 70, 14, 42, 26, 74, 64, 12, 42, 14

Offset: 1

Benoit Cloitre, Mar 10 2002

nonn

G. C. Greubel, Table of n, a(n) for n = 1..5000 (terms 1..300 from Vincenzo Librandi)
Wikipedia, Factorial number system
Index entries for factorial base representation of noninteger constants

Cf. A075874 (same for Pi), A007514 (different variant).

Cf. A067279 (zeta(2)), A067277 (zeta(3)), A068447 (zeta(4)), A068455 (zeta(6)), A068456 (zeta(7)), A068457 (zeta(8)), A068458 (zeta(9)), A068459 (zeta(10)).

SetDefaultRealField(RealField(250)); b:=Evaluate(RiemannZeta(),5); [n eq 1 select Floor(b) else Floor(Factorial(n)*b) - n*Floor(Factorial(n)*b/n) : n in [1..100]]; // G. C. Greubel, Nov 26 2018

t = Zeta[5]; s = {}; Do[n = Floor[t*i!]; t -= n/i!; AppendTo[s, n], {i, 1, 30}]; s (* Amiram Eldar, Nov 25 2018 *)
With[{b = Zeta[5]}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]] (* G. C. Greubel, Nov 26 2018 *)

vector(N=100, n, if(n>1, c=c%1*n, c=zeta(precision(5.,N*log(N/2.7)\2.3+3)))\1) \\ Specific a(n) can be computed via the FORMULA. For repeated use the value of c can be stored as a global variable, to be re-computed with higher precision when log_10(n!) exceeds its precision. - M. F. Hasler, Nov 25 2018

b=zeta(5)
@cached_function
def A068454(n):
    if n == 1: return floor(b)
    else: return expand(floor(factorial(n)*b) - n*floor(factorial(n-1)*b))
[A068454(n) for n in (1..100)] # G. C. Greubel, Nov 26 2018

a(n) = floor(c*n!) - n*floor(c*(n-1)!) = floor(frac(c*(n-1)!)*n) for n > 1, with c = zeta(5). - M. F. Hasler, Dec 20 2018

Name edited and keyword cons removed by M. F. Hasler, Nov 25 2018

cp's OEIS Frontend

A052277 a(n) = (4n+2)!/2^(2n+1).

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A333972 Decimal expansion of Pi^6/540 = zeta(2) * zeta(4).

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A067277 Factorial expansion of zeta(3): zeta(3) = Sum_{n>=1} a(n)/n!.

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A067279 Factorial expansion of zeta(2) : zeta(2) = Sum_{n>=1} a(n)/n!.

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A068454 Factorial expansion of zeta(5) = Sum_{n>=1} a(n)/n!, with a(n) as large as possible.

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Programs

Magma

Mathematica

PARI

Sage

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Extensions