A128507 -id:A128507 - 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

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

A128506 Numerators of partial sums for a series for 3sqrt(2)(Pi^3)/2^7.

Original entry on oeis.org

1, 28, 3473, 1187864, 32115203, 42776591068, 93938569006771, 93911487925744, 461478538827646397, 3165730339378740709148, 452199680641199918039, 5501473517781557885536888, 687727017229797976494536483

Offset: 0

Wolfdieter Lang, Apr 04 2007

The denominators are given in A128507.
The limit n -> infinity of the rationals r(n) defined below is 3*sqrt(2)*(Pi^3)/2^7 = 1.027756...
This series is obtained from the Fourier series for y(x)= x*(Pi-x) if 0<=x<=Pi and y(x)= (Pi-x)*(2*Pi-x) if Pi<=x<=2*Pi evaluated at x=Pi/4.

			Rationals r(n): [1, 28/27, 3473/3375, 1187864/1157625, 32115203/31255875,...].
3*sqrt(2)*(Pi^3)/2^7 = 1/1^3 + 1/3^3 - 1/5^3 - 1/7^3 + 1/9^3 + 1/11^3 - 1/13^3 - 1/15^3 + ...

G. C. Greubel, Table of n, a(n) for n = 0..385
W. Lang, Rationals and limit.

r[n_]:= Sum[(1/2)*((1-I)*I^k+(1+I)*(-I)^k)/(2k+1)^3, {k,0,n}]; Numerator[Table[r[n], {n,0,30}]] (* G. C. Greubel, Mar 28 2018 *)

{r(n) = sum(k=0,n, ((1-I)*I^k + (1+I)*(-I)^k)/(2*(2*k+1)^3))};
for(n=0,30, print1(numerator(r(n)), ", ")) \\ G. C. Greubel, Mar 28 2018

a(n) = numerator(r(n)) with the rationals r(n) = Sum_{k=0..n} S(2*k,sqrt(2))/(2*k+1)^3 with Chebyshev's S-Polynomials S(2*k,sqrt(2))=[1,1,-1,-1] periodic sequence with period 4. See A057077.

A370691 Square array read by upward antidiagonals: T(n, k) = denominator( 2k!(-2)^kSum_{m=1..n}( 1/(2m-1)^(k+1) ) ).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 15, 9, 1, 1, 105, 225, 27, 1, 765765, 405810405, 91398648466125, 48049812916875, 1033788065625, 89339709375, 3796875, 729, 1, 1, 1, 315, 11025, 3375, 27, 1, 1, 3465, 99225, 1157625, 16875, 81, 1, 1, 45045, 12006225, 31255875, 40516875, 253125, 243, 1, 1, 45045, 2029052025

Offset: 0

Thomas Scheuerle, Apr 21 2024

			array begins:
1,    1,        1,           1,              1,                  1
1,    1,        1,           1,              1,                  1
3,    9,        27,          27,             81,                 243
15,   225,      3375,        16875,          253125,             759375
105,  11025,    1157625,     40516875,       4254271875,         89339709375
315,  99225,    31255875,    3281866875,     1033788065625,      65128648134375
3465, 12006225, 41601569625, 48049812916875, 166492601756971875, 115379373017581509375

Cf. A370692 (numerators),
Cf. A025547 (first column), A128492 (second column).
Cf. A128507.
Cf. A255008 (denominators polygamma(n, 1) - polygamma(n, k)).
Cf. A255009 (numerators polygamma(n, 1) - polygamma(n, k)).

A := (n, k) -> Psi(k, n + 1/2) - Psi(k, 1/2):
seq(lprint(seq(denom(A(n, k)), k = 0..4)), n=0..6);

T(n, k) = denominator(sum(m=1, n, 1/(2*m-1)^(k+1))*k!*(-2)^k*2)

T(n, k) = denominator( polygamma(k, n + 1/2) - polygamma(k, 1/2) ).
T(n, k) = denominator( k!*(-1)^(k+1)*(zeta((k+1), 1/2 + n) - zeta((k+1), 1/2)) ), where zeta is the Hurwitz zeta function.
T(n, 0) = A025547(n).
T(n, 1) = A128492(n).
Conjectured: T(n, 2) = A128507(n).

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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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A128506 Numerators of partial sums for a series for 3*sqrt(2)*(Pi^3)/2^7.

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A370691 Square array read by upward antidiagonals: T(n, k) = denominator( 2*k!*(-2)^k*Sum_{m=1..n}( 1/(2*m-1)^(k+1) ) ).

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A128506 Numerators of partial sums for a series for 3sqrt(2)(Pi^3)/2^7.

A370691 Square array read by upward antidiagonals: T(n, k) = denominator( 2k!(-2)^kSum_{m=1..n}( 1/(2m-1)^(k+1) ) ).