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A013662 Decimal expansion of zeta(4). 17
1, 0, 8, 2, 3, 2, 3, 2, 3, 3, 7, 1, 1, 1, 3, 8, 1, 9, 1, 5, 1, 6, 0, 0, 3, 6, 9, 6, 5, 4, 1, 1, 6, 7, 9, 0, 2, 7, 7, 4, 7, 5, 0, 9, 5, 1, 9, 1, 8, 7, 2, 6, 9, 0, 7, 6, 8, 2, 9, 7, 6, 2, 1, 5, 4, 4, 4, 1, 2, 0, 6, 1, 6, 1, 8, 6, 9, 6, 8, 8, 4, 6, 5, 5, 6, 9, 0, 9, 6, 3, 5, 9, 4, 1, 6, 9, 9, 9, 1 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

zeta(4) = Pi^4/90 = 1.0823232337111381915160036965411679027747... [Harry J. Smith, Apr 29 2009]

REFERENCES

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 811.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 89, Exercise.

LINKS

Harry J. Smith, Table of n, a(n) for n=1,...,20000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Simon Plouffe, Pi^4/90 to 100000 digits

Simon Plouffe, Zeta(4) or Pi^4/90 to 10000 places

_Peter Bala_, New series for old functions

D. H. Bailey, J. M. Borwein and D. M. Bradley, Experimental determination of Ap'ery-like identities for zeta(4n+2)

D. Borwein and J. M. Borwein, On an intriguing integral and some series related to zeta(4) Proc. Amer. Math. Soc., Vol. 123, No.4, April 1995.

J. M. Borwein, D. J. Broadhurst, J. Kamnitzer, Central binomial sums, multiple Clausen values and zeta values arXiv:hep-th/0004153v1

L. Euler, On the sums of series of reciprocals

L. Euler, De summis serierum reciprocarum, E41.

R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, Arxiv preprint arXiv:1202.3670, 2012 - From N. J. A. Sloane, Jun 13 2012

FORMULA

From Peter Bala, Dec 03 2013: (Start)

Definition: zeta(4) := sum {n >= 1} 1/n^4.

Zeta(4) = 4/17*sum {n >= 1} ( (1 + 1/2 + ... + 1/n)/n )^2 and

Zeta(4) = 16/45*sum {n >= 1} ( (1 + 1/3 + ... + 1/(2*n-1))/n )^2 (see Borwein and Borwein).

zeta(4) = 256/90*sum {n >= 1} n^2*(4*n^2 + 3)*(12*n^2 + 1)/(4*n^2 - 1)^5.

Series acceleration formulas:

zeta(4) = 36/17*sum {n >= 1} 1/( n^4*binomial(2*n,n) ) (Comtet)

= 36/17*sum {n >= 1} P(n)/( (2*n(2*n - 1))^4*binomial(4*n,2*n) )

= 36/17*sum {n >= 1} Q(n)/( (3*n(3*n - 1)*(3*n - 2))^4*binomial(6*n,3*n) ),

where P(n) = 80*n^4 - 48*n^3 + 24*n^2 - 8*n + 1 and Q(n) = 137781*n^8 - 275562*n^7 + 240570*n^6 - 122472*n^5 + 41877*n^4 - 10908*n^3 + 2232*n^2 - 288*n + 16 (see section 8 in the Bala link).(End)

zeta(4) = 2/3*2^4/(2^4 - 1)*( sum {n even} n^2*p(n)/(n^2 - 1)^5 ), where p(n) = 3*n^4 + 10*n^2 + 3 is a row polynomial of A091043. See A013664, A013666, A013668 and A013670. - Peter Bala, Dec 05 2013

MATHEMATICA

RealDigits[Zeta[4], 10, 120][[1]] (* Harvey P. Dale, Dec 18 2012 *)

PROG

(PARI) { default(realprecision, 20080); x=Pi^4/90; for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b013662.txt", n, " ", d)); } \\ Harry J. Smith, Apr 29 2009

(Maxima) ev(zeta(4), numer) ; /* R. J. Mathar, Feb 27 2012 */

CROSSREFS

Cf. A013664, A013666, A013668, A013670.

Sequence in context: A021928 A185111 A086058 * A222225 A140244 A160105

Adjacent sequences:  A013659 A013660 A013661 * A013663 A013664 A013665

KEYWORD

nonn,cons

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified August 2 04:37 EDT 2014. Contains 245138 sequences.