A164656 -id:A164656 - cp's OEIS frontend

A164655 Numerators of partial sums of Theta(3) = Sum_{j>=1} 1/(2*j-1)^3.

1, 28, 3527, 1213136, 32797547, 43684790932, 96017087247229, 96044168328256, 471956397645187853, 3237597973008257555852, 462561506842656976961, 5628425850334528955928112, 703596058798919360293439483, 18998011529681231695738912916, 463360571051954739540899597748949

Offset: 1

Wolfdieter Lang, Oct 16 2009

Warning: Usually, Theta3(x) = Sum_{n=-oo..+oo} x^(n^2). - Joerg Arndt, Mar 31 2024
The denominators look like those given for the partial sums of another series in A128507.
Rationals (partial sums) Theta(3,n) := Sum_{j=1..n} 1/(2*j-1)^3 (in lowest terms). The limit of these rationals is Theta(3) = (1-1/2^3)*Zeta(3) approximately 1.051799790 (Zeta(n) is the Euler-Riemann zeta function).
This is a member of the k-family of rational sequences Theta(k,n) := Sum_{j=1..n} 1/(2*j-1)^k, k >= 1, which coincides for k=1 with A025550/A025547 (but only for the first 38 terms), for k=2 with A120268/A128492, for k=3 with a(n)/A128507(n) (the denominators may depart for higher n values), A120269/A128493 and A164656/A164657, for k=4 and 5, respectively.

			Rationals Theta(3,n): [1, 28/27, 3527/3375, 1213136/1157625, 32797547/31255875, 43684790932/41601569625, ...].

R. Ayoub, Euler and the Zeta Function, Am. Math. Monthly 81 (1974) 1067-1086, p. 1070.
Wolfdieter Lang, Theta(k, n), k-family of rational sequences and limits.

r[n_] := Sum[1/(2*j-1)^3, {j, 1, n}]; (* or r[n_] := (PolyGamma[2, n+1/2] - PolyGamma[2, 1/2])/16 // FullSimplify; *) Table[r[n] // Numerator, {n, 1, 15}] (* Jean-François Alcover, Dec 02 2013 *)

a(n) = numerator(Theta(3,n)) = numerator(Sum_{j=1..n} 1/(2*j-1)^3), n >= 1.

Theta(3,n) = (-Psi(2, 1/2) + Psi(2, n+1/2))/16, n >= 1, where Psi(n, k) = Polygamma(n,k) is the n-th derivative of the digamma function. Psi(2, 1/2) = -14*Zeta(3). - Jean-François Alcover, Dec 02 2013

A164657 Denominators of partial sums of Theta(5) = Sum_{j>=1} 1/(2*j-1)^5.

Original entry on oeis.org

1, 243, 759375, 12762815625, 3101364196875, 499477805270915625, 185452612752454075153125, 185452612752454075153125, 263316190384861185784690603125, 651996955695764397260286617707209375, 651996955695764397260286617707209375, 4196476041813743307955464949873473110315625

Offset: 1

Wolfdieter Lang, Oct 16 2009

The numerators are given by A164656.
For a reference and a W. Lang link see A164656.
Rationals (partial sums) Theta(5,n) := Sum_{j=1..n} 1/(2*j-1)^5 (in lowest terms). The limit of these rationals is Theta(5)= (1-1/2^5)*Zeta(5) approximately 1.004523763 (Zeta(n) is the Euler-Riemann zeta function).

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

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] // Denominator, {n, 1, 12}] (* Jean-François Alcover, Dec 02 2013 *)

a(n) = denominator(Theta(5,n)) = denominator(Sum_{j=1..n} 1/(2*j-1)^5).

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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A164655 Numerators of partial sums of Theta(3) = Sum_{j>=1} 1/(2*j-1)^3.

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A164657 Denominators of partial sums of Theta(5) = Sum_{j>=1} 1/(2*j-1)^5.

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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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