A173947 -id:A173947 - cp's OEIS frontend

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

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

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

A360945 a(n) = numerator of (Zeta(2n+1,1/4) - Zeta(2n+1,3/4))/Pi^(2*n+1) where Zeta is the Hurwitz zeta function.

Original entry on oeis.org

1, 2, 10, 244, 554, 202084, 2162212, 1594887848, 7756604858, 9619518701764, 59259390118004, 554790995145103208, 954740563911205348, 32696580074344991138888, 105453443486621462355224, 7064702291984369672858925136, 4176926860695042104392112698

Offset: 0

Artur Jasinski, Feb 26 2023

The function (Zeta(2*n+1,1/4) - Zeta(2*n+1,3/4))/Pi^(2*n+1) is rational for every positive integer n.
For denominators see A360966.
(Zeta(2*n+1,1/4) + Zeta(2*n+1,3/4))/Zeta(2*n+1) = 4*16^n - 2*4^n; see A193475.
For numerators of the function (Zeta(2*n,1/4) + Zeta(2*n,3/4))/Pi^(2*n) see A361007.
For denominators of the function (Zeta(2*n,1/4) + Zeta(2*n,3/4))/Pi^(2*n) see A036279.
(Zeta(2*n,1/4) - Zeta(2*n,3/4))/beta(2*n) = 16^n (see A001025) where beta is the Dirichlet beta function.
From the above formulas we can express Zeta(k,1/4) and Zeta(k,3/4) for every positive integer k.

			a(0) = 1 because lim_{n->0} (Zeta(2*n+1,1/4) - Zeta(2*n+1,3/4))/Pi^(2*n+1) = 1.
a(3) = 244 because (Zeta(2*3+1,1/4) - Zeta(2*3+1,3/4))/Pi^(2*3+1) = 244/45.

Cf. A000364, A046982, A173945, A173947, A173948, A173949, A173953, A173954, A173955, A173982, A173983, A173984, A173987, A360966, A361007, A361007.

Table[(Zeta[2*n + 1, 1/4] - Zeta[2*n + 1, 3/4]) / Pi^(2*n + 1), {n, 1, 25}] // FunctionExpand // Numerator (* Vaclav Kotesovec, Feb 27 2023 *)
t[0, 1] = 1; t[0, _] = 0;
t[n_, k_] := t[n, k] = (k-1) t[n-1, k-1] + (k+1) t[n-1, k+1];
a[n_] := Sum[t[2n, k]/(2n)!, {k, 0, 2n+1}] // Numerator;
Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Mar 15 2023 *)
a[n_] := SeriesCoefficient[Tan[x+Pi/4], {x, 0, 2n}] // Numerator;
Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Apr 15 2023 *)

a(n) = numerator(abs(eulerfrac(2*n))*(2*n + 1)*2^(2*n)/(2*n + 1)!); \\ Michel Marcus, Apr 11 2023

a(n) = A046982(2*n).

(Zeta(2*n + 1, 1/4) - Zeta(2*n + 1, 3/4))/(Pi^(2*n + 1)) = A000364(n)*(2*n + 1)*2^(2*n)/(2*n + 1)!.

A360966 a(n) = denominator of (Zeta(2n+1,1/4) - Zeta(2n+1,3/4))/Pi^(2*n+1) where Zeta is the Hurwitz zeta function.

Original entry on oeis.org

1, 1, 3, 45, 63, 14175, 93555, 42567525, 127702575, 97692469875, 371231385525, 2143861251406875, 2275791174570375, 48076088562799171875, 95646113035463615625, 3952575621190533915703125, 1441527579493018251609375, 68739242628124575327993046875, 333120945043988326589504765625

Offset: 0

Artur Jasinski, Apr 09 2023

The function (Zeta(2*n+1,1/4) - Zeta(2*n+1,3/4))/Pi^(2*n+1) is rational for every positive integer n.

For numerators see A360945.

			a(0) = 1 because lim_{n->0} (Zeta(2*n+1,1/4) - Zeta(2*n+1,3/4))/Pi^(2*n+1) = 1.
a(3) = 45 because (Zeta(2*3+1,1/4) - Zeta(2*3+1,3/4))/Pi^(2*3+1) = 244/45.

Wikipedia, Hurwitz zeta function

Bisection of A046983.
Cf. A360945 (numerators).
Cf. A000364, A046982, A173945, A173947, A173948, A173949, A173953, A173954, A173955, A173982, A173983, A173984, A173987, A361007.

Table[(Zeta[2*n + 1, 1/4] - Zeta[2*n + 1, 3/4]) / Pi^(2*n + 1), {n, 0, 25}] // FunctionExpand // Denominator
(* Second program: *)
a[n_] := SeriesCoefficient[Tan[x + Pi/4], {x, 0, 2n}] // Denominator;
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Apr 16 2023 *)

a(n) = denominator(abs(eulerfrac(2*n))*(2*n + 1)*2^(2*n)/(2*n + 1)!); \\ Michel Marcus, Apr 11 2023

a(n) = A046983(2*n).

(Zeta(2*n + 1, 1/4) - Zeta(2*n + 1, 3/4))/(Pi^(2*n + 1)) = A000364(n)*(2*n + 1)*2^(2*n)/(2*n + 1)!.

A361007 a(n) = numerator of (zeta(2n,1/4) + zeta(2n,3/4))/Pi^(2*n) where zeta is the Hurwitz zeta function.

Original entry on oeis.org

0, 2, 8, 64, 2176, 31744, 2830336, 178946048, 30460116992, 839461371904, 232711080902656, 39611984424992768, 955693069653508096, 1975371841521663868928, 1124025625663103358205952, 369906947004953565463576576, 278846808228005417477465964544

Offset: 0

Artur Jasinski, Mar 15 2023

The function (zeta(2*n,1/4) + zeta(2*n,3/4))/Pi^(2*n) is rational for every positive integer n.

			tan(2*x) = 2*x + (8/3)*x^3 + (64/15)*x^5 + (2176/315)*x^7 + (31744/2835)*x^9 + ...

Cf. A036279 (denominators).

Cf. A000364, A046982, A173945, A173947, A173948, A173949, A173953, A173954, A173955, A173982, A173983, A173984, A173987, A360945, A360966.

Table[(Zeta[2*n, 1/4] + Zeta[2*n, 3/4])/Pi^(2*n), {n, 0, 25}] //
  FunctionExpand // Numerator
Table[4^(2 k) (2^(2 k) - 1) (-1)^(k + 1) BernoulliB[2 k]/(2 (2 k)!), {k, 0, 25}] // Numerator

my(x='x+O('x^50), v = Vec(tan(2*x)/x)); apply(numerator, vector(#v\2, k, v[2*k-1])) \\ Michel Marcus, Apr 09 2023

a(n) = numerator( [x^(2*n-1)] tan(2*x) ).

a(n) = numerator( (-1)^(n + 1)*4^(2*n)*(2^(2*n) - 1)*B(2*n)/(2*(2*n)!) ) where B(2*n) are Bernoulli numbers.

cp's OEIS Frontend

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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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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A360945 a(n) = numerator of (Zeta(2*n+1,1/4) - Zeta(2*n+1,3/4))/Pi^(2*n+1) where Zeta is the Hurwitz zeta function.

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A360966 a(n) = denominator of (Zeta(2*n+1,1/4) - Zeta(2*n+1,3/4))/Pi^(2*n+1) where Zeta is the Hurwitz zeta function.

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A361007 a(n) = numerator of (zeta(2*n,1/4) + zeta(2*n,3/4))/Pi^(2*n) where zeta is the Hurwitz zeta function.

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Formula

A360945 a(n) = numerator of (Zeta(2n+1,1/4) - Zeta(2n+1,3/4))/Pi^(2*n+1) where Zeta is the Hurwitz zeta function.

A360966 a(n) = denominator of (Zeta(2n+1,1/4) - Zeta(2n+1,3/4))/Pi^(2*n+1) where Zeta is the Hurwitz zeta function.

A361007 a(n) = numerator of (zeta(2n,1/4) + zeta(2n,3/4))/Pi^(2*n) where zeta is the Hurwitz zeta function.