A164656 Numerators of partial sums of Theta(5) = sum( 1/(2*j-1)^5, j=1..infinity ).

1, 244, 762743, 12820180976, 3115356499043, 501734380891571068, 186290962962179367466549, 186291207179611798681792, 264507060005034822095008296869, 654945930087597102815813733559637156, 654946089730308117005814730177159031, 4215458332009996232497953858159263996273008

Offset

1,2

Comments

The denominators are given by A164657.

Rationals (partial sums) Theta(5,n) := sum(1/(2*j-1)^5,j=1..n) (in lowest terms). The limit of these rationals is Theta(5)= (1-1/2^5)*Zeta(5) approximately 1.004523763.., see A013663.

This is a member of the k-family of rational sequences Theta(k,n):=sum(1/(2*j-1)^k,j=1..n), k>=1, which includes A025550/A025547 (but only for the first 38 entries), A120268/A128492, A164655(n)/A128507(n) (the denominators may depart for higher n values), A120269/A128493, a(n)/A164657, for k=1..5.

Links

Table of n, a(n) for n=1..12.

R. Ayoub, Euler and the Zeta Function, Am. Math. Monthly 81 (1974) 1067-1086.

W. Lang: Theta(k,n), k-family of rational sequences and limits.

Formula

a(n) = numer(Theta(5,n))= numerator(sum(1/(2*j-1)^5,j=1..n)), n>=1.

Theta(5,n) = (-Psi(4, 1/2) + Psi(4, n+1/2))/(4!*2^5), n >= 1, with Psi(n,k) = Polygamma(n,k) is the n^th derivative of the digamma function. Psi(4, 1/2) = -4!*31*Zeta(5). - Jean-François Alcover, Dec 02 2013

Example

Rationals Theta(5,n): [1, 244/243, 762743/759375, 12820180976/12762815625, 3115356499043/3101364196875,...].

Mathematica

r[n_] := Sum[1/(2*j-1)^5, {j, 1, n}]; (* or r[n_] := (PolyGamma[4, n+1/2] - PolyGamma[4, 1/2])/768 // FullSimplify; *) Table[r[n] // Numerator, {n, 1, 12}] (* Jean-François Alcover, Dec 02 2013 *)

Crossrefs

Sequence in context: A146552 A263948 A306981 * A291587 A220090 A013684

Adjacent sequences: A164653 A164654 A164655 * A164657 A164658 A164659

Keyword

nonn,frac,easy

Author

Wolfdieter Lang, Oct 16 2009

Status

approved