A120269 Numerator of Sum_{k=1..n} 1/(2k-1)^4.

1, 82, 51331, 123296356, 9988505461, 146251554055126, 4177234784807204311, 4177316109293528392, 348897735816424941428857, 45469045689642442391390873722, 45469276109166591994111574347

Offset: 1

Author

Alexander Adamchuk, Jul 01 2006

Keywords

• nonn
• frac

Comments

a((p-1)/2) is divisible by prime p > 5.

Denominators are in A128493.

The limit of the rationals r(n) = Sum_{k=1..n} 1/(2k-1)^4, for n -> infinity, is (Pi^4)/96 = (1 - 1/2^4)*zeta(4), which is approximately 1.014678032.

r(n) = (Psi(3, 1/2) - Psi(3, n+1/2))/(3!*2^4) for n >= 1, where Psi(n,k) = Polygamma(n,k) is the n^th derivative of the digamma function. Psi(3, 1/2) = 3!*15*zeta(4) = Pi^4. - Jean-François Alcover, Dec 02 2013

Links

• G. C. Greubel, Table of n, a(n) for n = 1..293
• W. Lang, Rationals and limit.

Crossrefs

Cf. A007410, A013662, A025550.

Programs

Magma

[Numerator((&+[1/(2*k-1)^4: k in [1..n]])): n in [1..20]]; // G. C. Greubel, Aug 23 2018

Mathematica

Numerator[Table[Sum[1/(2k-1)^4,{k,1,n}],{n,1,20}]]
Table[(PolyGamma[3, 1/2] - PolyGamma[3, n + 1/2])/(3!*2^4) // Simplify // Numerator, {n, 1, 15}] (* Jean-François Alcover, Dec 02 2013 *)

PARI

for(n=1,20, print1(numerator(sum(k=1,n, 1/(2*k-1)^4)), ", ")) \\ G. C. Greubel, Aug 23 2018