A164655 -id:A164655 - cp's OEIS frontend

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

Wolfdieter Lang, Oct 16 2009

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.

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

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.

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 *)

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

A234144 a(n) = numerator of Sum_(k=1..n) 1/(2*k - 1)^n.

Original entry on oeis.org

0, 1, 10, 3527, 123296356, 3115356499043, 1733194364791766081374, 376470435881775086250915790503469, 16952748458548438370767527584555153032, 90548635884513844033505877600764150558334149264809109

Offset: 0

Jean-François Alcover, Dec 20 2013

The sequence A234144(n)/A234145(n) is Theta(n, n), as defined by Wolfdieter Lang.

Robert Israel, Table of n, a(n) for n = 0..35
Wolfdieter Lang, Theta(k, n), k-family of rational sequences and limits.

Cf. A164655, A164656, A234145 (denominators).

f:= proc(n) local k; numer(add(1/(2*k-1)^n,k=1..n)); end proc:
map(f, [$0..10]); # Robert Israel, Nov 05 2023

a[n_] := Sum[1/(2*k-1)^n, {k, 1, n}] // Numerator; Table[a[n], {n, 0, 10}]

a(n) = numerator of (2^n*Zeta(n) - Zeta(n) - Zeta(n, n+1/2))/2^n.
a(n) = numerator of ((-1/2)^n*(PolyGamma(n-1, 1/2) - PolyGamma(n-1, n+1/2)))/(n-1)!.
A234144(n) / A234145(n) ~ 1.

A234145 a(n) = denominator of sum_(k=1..n) 1/(2*k-1)^n.

Original entry on oeis.org

1, 1, 9, 3375, 121550625, 3101364196875, 1730690595263722640625, 376292999446907764908950466328125, 16950118160085960270323673755750390625, 90543986887356385297750500755391437150880164126953125

Offset: 0

Jean-François Alcover, Dec 20 2013

The sequence A234144(n)/A234145(n) is Theta(n, n), as defined by Wolfdieter Lang.

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

Cf. A164655, A164656, A234144 (numerators).

a[n_] := Sum[1/(2*k-1)^n, {k, 1, n}] // Denominator; Table[a[n], {n, 0, 10}]

a(n) = denominator of (2^n*Zeta(n) - Zeta(n) - Zeta(n, n+1/2))/2^n.
a(n) = denominator of ((-1/2)^n*(PolyGamma(n-1, 1/2) - PolyGamma(n-1, n+1/2)))/(n-1)!.
A234144(n) / A234145(n) ~ 1.

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

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A234144 a(n) = numerator of Sum_(k=1..n) 1/(2*k - 1)^n.

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A234145 a(n) = denominator of sum_(k=1..n) 1/(2*k-1)^n.

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