A120268 -id:A120268 - cp's OEIS frontend

A173987 a(n) = denominator of ((Zeta(0,2,2/3) - Zeta(0,2,n+2/3))/9), where Zeta is the Hurwitz Zeta function.

1, 4, 100, 1600, 193600, 9486400, 2741569600, 2741569600, 1450290318400, 245099063809600, 206128312663873600, 3298053002621977600, 3298053002621977600, 1190597133946533913600, 2001393782164123508761600

Offset: 0

Artur Jasinski, Mar 04 2010

G. C. Greubel, Table of n, a(n) for n = 0..300

For numerators see A173985.

Cf. A006752, A120268, A173945, A173947, A173948, A173949, A173953, A173955, A173973, A173982, A173983, A173984, A173986.

[1] cat [Denominator((&+[9/(3*k+2)^2: k in [0..n-1]])): n in [1..20]]; // G. C. Greubel, Aug 23 2018

a := n -> (Zeta(0,2,2/3) - Zeta(0,2,n+2/3))/9:
seq(denom(a(n)), n=0..14); # Peter Luschny, Nov 14 2017

Table[FunctionExpand[(1/9)*(4*(Pi^2)/3 - Zeta[2, 1/3] - Zeta[2, (3*n + 2)/3])], {n, 0, 20}] // Denominator (* Vaclav Kotesovec, Nov 13 2017 *)
Denominator[Table[Sum[9/(3*k + 2)^2, {k, 0, n - 1}], {n, 0, 20}]] (* G. C. Greubel, Aug 23 2018 *)

for(n=0,20, print1(denominator(9*sum(k=0,n-1, 1/(3*k+2)^2)), ", ")) \\ G. C. Greubel, Aug 23 2018

a(n) = denominator of 2*(Pi^2)/3 - J - Zeta(2,(3*n+2)/3), where Zeta is the Hurwitz Zeta function and J is the constant A173973.

a(n) = denominator of Sum_{k=0..(n-1)} 9/(3*k+2)^2. - G. C. Greubel, Aug 23 2018

Name simplified by Peter Luschny, Nov 14 2017

A173984 a(n) is the denominator of (Zeta(0,2,1/3) - Zeta(0,2,n+1/3)) where Zeta is the Hurwitz Zeta function.

Original entry on oeis.org

1, 1, 16, 784, 19600, 3312400, 52998400, 19132422400, 2315023110400, 57875577760000, 57875577760000, 55618430227360000, 16073726335707040000, 22004931353582937760000, 22004931353582937760000

Offset: 0

Artur Jasinski, Mar 04 2010

G. C. Greubel, Table of n, a(n) for n = 0..300

For the numerators see A173982.

Cf. A006752, A120268, A173945, A173947, A173948, A173949, A173953, A173955, A173973, A173983, A173985, A173986, A173987.

[1,1] cat [Denominator((&+[9/(3*k+1)^2: k in [1..n-1]])): n in [2..20]]; // G. C. Greubel, Aug 24 2018

a := n -> Zeta(0,2,1/3) - Zeta(0,2,n+1/3):
seq(denom(a(n)), n=0..14); # Peter Luschny, Nov 14 2017

Table[FunctionExpand[-Zeta[2, (3*n + 1)/3] + Zeta[2, 1/3]], {n, 0, 20}] // Denominator (* Vaclav Kotesovec, Nov 13 2017 *)
Denominator[Table[Sum[9/(3*k + 1)^2, {k, 1, n - 1}], {n, 0, 30}]] (* G. C. Greubel, Aug 24 2018 *)

for(n=0,20, print1(denominator(sum(k=1,n-1, 9/(3*k+1)^2)), ", ")) \\ G. C. Greubel, Aug 24 2018

a(n) = denominator of 2*(Pi^2)/3 + J - Zeta(2,(3*n+1)/3), where Zeta is the Hurwitz Zeta function and the constant J is A173973.

a(n) = denominator of Sum_{k=1..(n-1)} 9/(3*k+1)^2. - G. C. Greubel, Aug 24 2018

Name simplified by Peter Luschny, Nov 14 2017

A173986 a(n) = numerator((Psi(1, 2/3) - Psi(1, n+2/3))/9), where Psi(1, z) is the Trigamma function.

Original entry on oeis.org

0, 1, 29, 489, 60769, 3026081, 884023809, 890877733, 474015890357, 80471258049933, 67921427083803253, 1089963588226225073, 1092655876391630769, 395273284628034202009, 665644988593672027490729

Offset: 0

Artur Jasinski, Mar 04 2010

a(n+1)/A173987(n+1) gives, for n >= 0, the partial sum Sum_{k=0..n} 1/(3*k+2)^2. The limit n -> infinity is given in A294967 as the Hurwitz Zeta function or the Trigamma function (1/9)*Zeta(2, 2/3) = (1/9)*Psi(1, 2/3) = 0.3404306010 ... - Wolfdieter Lang, Nov 12 2017

			The rationals a(n)/A173987(n) begin 0/1, 1/4, 29/100, 489/1600, 60769/193600, 3026081/9486400, 884023809/2741569600, 890877733/2741569600, ... - _Wolfdieter Lang_, Nov 12 2017

G. C. Greubel, Table of n, a(n) for n = 0..300

For denominators see A173987.
For 9*A173986 see A173985.
Cf. A006752, A120268, A173945, A173947, A173948, A173949, A173953, A173955, A173973, A173982, A173983, A173984, A173985, A294967.

[0] cat [Numerator((&+[2/(3*k+2)^2: k in [0..n-2]])): n in [2..20]]; // G. C. Greubel, Aug 23 2018

r := n -> (Psi(1, 2/3) - Psi(1, n+2/3))/9:
seq(numer(simplify(r(n))), n=0..14); # Peter Luschny, Nov 13 2017

Table[Numerator[FunctionExpand[(4*Pi^2/3 - Zeta[2, 1/3] - Zeta[2, (3*n + 2)/3])/9]], {n, 0, 20}] (* Vaclav Kotesovec, Nov 14 2017 *)
Numerator[Table[Sum[2/(3*k + 2)^2, {k, 0, n - 2}], {n, 1, 20}]] (* G. C. Greubel, Aug 23 2018 *)

for(n=1,20, print1(numerator(sum(k=0,n-2, 2/(3*k+2)^2)), ", ")) \\ G. C. Greubel, Aug 23 2018

a(n) = numerator(r(n)) with r(n) = (1/9)*(4*(Pi^2)/3 - Zeta(2, 1/3) - Zeta(2, (3*n+2)/3)) = (1/9)*(Zeta(2, 2/3) - Zeta(2, (3*n+2)/3)) with the Hurwitz Zeta function Zeta(2, q). This becomes the formula given in the name. - Wolfdieter Lang, Nov 13 2017
a(n) = numerator of (1/9)*(2(Pi^2)/3 - J - Zeta(2, (3n+2)/3)) where J is the constant A173973 [which becomes the preceding formula].
a(n) = numerator of Sum_{k=0..(n-2)} 2/(3*k+2)^2. - G. C. Greubel, Aug 23 2018

Name simplified by Peter Luschny, Nov 13 2017

A128492 Denominator of Sum_{k=1..n} 1/(2*k-1)^2.

Original entry on oeis.org

1, 9, 225, 11025, 99225, 12006225, 2029052025, 405810405, 117279207045, 42337793743245, 42337793743245, 22396692890176605, 2799586611272075625, 25196279501448680625, 21190071060718340405625

Offset: 1

Wolfdieter Lang, Apr 04 2007

Old definition was "Denominators of partial sums for a series for (Pi^2)/8".

See the comments and the Wolfdieter Lang link.

			Fractions begin: 1, 10/9, 259/225, 12916/11025, 117469/99225, 14312974/12006225, 2430898831/2029052025, 487983368/405810405, ... = A120268/A128492.

Wolfdieter Lang, Rationals and limit.

Cf. A120268 (numerators).

a[n_] := Pi^2/8 - PolyGamma[1, n+1/2]/4 // Simplify // Denominator; Table[a[n], {n, 1, 15}] (* Jean-François Alcover, Dec 17 2013 *)

a(n) = denominator(sum(k=1, n, 1/(2*k-1)^2)); \\ Michel Marcus, May 09 2020

a(n) = denominator( Pi^2/2 - Zeta(2,(2*n+1)/2) ) for n > 0; see Artur Jasinski in A120268. - Bruno Berselli, Dec 02 2013

Also equals denominator( Pi^2/8 - PolyGamma(1, n+1/2)/4 ). - Jean-François Alcover, Dec 17 2013

Definition replaced with Lang's formula by Bruno Berselli, Dec 02 2013

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

Original entry on oeis.org

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

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

Original entry on oeis.org

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

A173985 a(n) = numerator of (Zeta(0,2,2/3) - Zeta(0,2,n+2/3)), where Zeta is the Hurwitz Zeta function.

Original entry on oeis.org

0, 9, 261, 4401, 546921, 27234729, 7956214281, 8017899597, 4266143013213, 724241322449397, 611292843754229277, 9809672294036025657, 9833902887524676921, 3557459561652307818081, 5990804897343048247416561

Offset: 0

Artur Jasinski, Mar 04 2010

All numbers in this sequence are divisible by 9.

G. C. Greubel, Table of n, a(n) for n = 0..300

For denominators see A173987.
For A173985/9 see A173986.
Cf. A006752, A120268, A173945, A173947, A173948, A173949, A173953, A173955, A173973, A173982, A173983, A173984.

[0] cat [Numerator((&+[9/(3*k+1)^2: k in [0..n-1]])): n in [1..20]]; // G. C. Greubel, Aug 23 2018

A173985 := proc(n) add( 1/(2/3+i)^2,i=0..n-1) ; numer(%) ; end proc: seq(A173985(n),n=0..20) ; # R. J. Mathar, Apr 22 2010

Table[FunctionExpand[4*(Pi^2)/3 - Zeta[2, 1/3] - Zeta[2, (3*n + 2)/3]], {n, 0, 20}] // Numerator (* Vaclav Kotesovec, Nov 13 2017 *)
Numerator[Table[Sum[9/(3*k + 1)^2, {k, 0, n - 1}], {n, 0, 20}]] (* G. C. Greubel, Aug 23 2018 *)

for(n=0,20, print1(numerator(9*sum(k=0,n-1, 1/(3*k+1)^2)), ", ")) \\ G. C. Greubel, Aug 23 2018

a(n) = numerator of 2*(Pi^2)/3 - J - Zeta(2, (3*n+2)/3), where Zeta is the Hurwitz Zeta function and the constant J is A173973.
a(n)/A173987(n) = sum_{i=0..n-1} 1/(i+2/3)^2 = psi'(2/3)-psi'(2/3+n). - R. J. Mathar, Apr 22 2010
a(n) = numerator of Sum_{k=0..(n-1)} 9/(3*k+1)^2. - G. C. Greubel, Aug 23 2018

Name simplified by Peter Luschny, Nov 14 2017

cp's OEIS Frontend

A173987 a(n) = denominator of ((Zeta(0,2,2/3) - Zeta(0,2,n+2/3))/9), where Zeta is the Hurwitz Zeta function.

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A173984 a(n) is the denominator of (Zeta(0,2,1/3) - Zeta(0,2,n+1/3)) where Zeta is the Hurwitz Zeta function.

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A173986 a(n) = numerator((Psi(1, 2/3) - Psi(1, n+2/3))/9), where Psi(1, z) is the Trigamma function.

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A128492 Denominator of Sum_{k=1..n} 1/(2*k-1)^2.

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

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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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A173985 a(n) = numerator of (Zeta(0,2,2/3) - Zeta(0,2,n+2/3)), where Zeta is the Hurwitz Zeta function.

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