A173948 -id:A173948 - cp's OEIS frontend

A003983 Array read by antidiagonals with T(n,k) = min(n,k).

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 3, 2, 1, 1, 2, 3, 4, 3, 2, 1, 1, 2, 3, 4, 4, 3, 2, 1, 1, 2, 3, 4, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 7, 7, 6, 5, 4, 3, 2, 1

Offset: 1

Marc LeBrun

Also, "correlation triangle" for the constant sequence 1. - Paul Barry, Jan 16 2006
Antidiagonal sums are in A002620.
As a triangle, row sums are A002620. T(2n,n)=n+1. Diagonal sums are A001399. Construction: Take antidiagonal triangle of MM^T where M is the sequence array for the constant sequence 1 (lower triangular matrix with all 1's). - Paul Barry, Jan 16 2006
From Franklin T. Adams-Watters, Sep 25 2011: (Start)
As a triangle, count up to ceiling(n/2) and back down again (repeating the central term when n is even).
When the first two instances of each number are removed from the sequence, the original sequence is recovered.
(End)

			Triangle version begins
  1;
  1, 1;
  1, 2, 1;
  1, 2, 2, 1;
  1, 2, 3, 2, 1;
  1, 2, 3, 3, 2, 1;
  1, 2, 3, 4, 3, 2, 1;
  1, 2, 3, 4, 4, 3, 2, 1;
  1, 2, 3, 4, 5, 4, 3, 2, 1;
  ...

Reinhard Zumkeller, Rows n=1..100 of triangle, flattened

Cf. A002620, A001399, A087062, A115236, A115237, A124258, A006752, A120268, A173945, A173947, A173948, A173949, A173953, A173954, A173955, A173973, A173982-A173986, A004197.

a003983 n k = a003983_tabl !! (n-1) !! (k-1)
a003983_tabl = map a003983_row [1..]
a003983_row n = hs ++ drop m (reverse hs)
   where hs = [1..n' + m]
         (n',m) = divMod n 2
-- Reinhard Zumkeller, Aug 14 2011

a(n) = min(floor(1/2 + sqrt(2*n)) - (2*n + round(sqrt(2*n)) - round(sqrt(2*n))^2)/2+1, (2*n + round(sqrt(2*n)) - round(sqrt(2*n))^2)/2) # Leonid Bedratyuk, Dec 13 2009

Flatten[Table[Min[n-k+1, k], {n, 1, 14}, {k, 1, n}]] (* Jean-François Alcover, Feb 23 2012 *)

T(n,k) = min(n,k) \\ Charles R Greathouse IV, Feb 06 2017

from math import isqrt
def A003983(n):
    a = (m:=isqrt(k:=n<<1))+(k>m*(m+1))
    x = n-(a*(a-1)>>1)
    return min(x,a-x+1) # Chai Wah Wu, Jun 14 2025

Number triangle T(n, k) = Sum_{j=0..n} [j<=k][j<=n-k]. - Paul Barry, Jan 16 2006
G.f.: 1/((1-x)*(1-x*y)*(1-x^2*y)). - Christian G. Bower, Jan 17 2006
a(n) = min(floor( 1/2 + sqrt(2*n)) - (2*n + round(sqrt(2*n)) - round(sqrt(2*n))^2)/2+1, (2*n + round(sqrt(2*n)) - round(sqrt(2*n))^2)/2). - Leonid Bedratyuk, Dec 13 2009

More terms from Larry Reeves (larryr(AT)acm.org), Nov 08 2000

Entry revised by N. J. A. Sloane, Dec 05 2006

A173947 a(n) = numerator of (Zeta(2, 1/4) - Zeta(2, n+1/4)), where Zeta is the Hurwitz Zeta function.

Original entry on oeis.org

0, 16, 416, 34096, 5794624, 1680121936, 82501802464, 2065646660464, 1739147340740224, 210617970218777104, 288533264855755545376, 485294472126860897387056, 485518650207447822251456

Offset: 0

Artur Jasinski, Mar 03 2010

For A173947/16 see A173949.

a(n+1)/A173948(n+1) =: r(n) = (Zeta(2, 1/4) - Zeta(2, n + 5/4)), the partial sum Sum_{k=0..n} 1/(k + 1/4)^2, n >= 0. The limit is Zeta(2, 1/4) = A282823 = 16*A222183. - Wolfdieter Lang, Nov 14 2017

G. C. Greubel, Table of n, a(n) for n = 0..250
Eric Weisstein's World of Mathematics, Hurwitz Zeta Function
Eric Weisstein's World of Mathematics, Trigamma Function

Cf. A006752, A120268, A173945, A173948 (denominators), A173949.

[1] cat [Numerator((&+[1/(4*k+1)^2: k in [0..n-1]])): n in [1..20]]; // G. C. Greubel, Aug 22 2018

r := n -> Psi(1, 1/4) - Zeta(0, 2, n+1/4):
seq(numer(simplify(r(n))), n=0..13); # Peter Luschny, Nov 14 2017

Table[Numerator[FunctionExpand[8*Catalan + Pi^2 - Zeta[2, (4*n + 1)/4]]], {n, 0, 20}] (* Vaclav Kotesovec, Nov 14 2017 *)
Numerator[Table[128*n*Sum[(1 + 4*k + 2*n) / ((1 + 4*k)^2*(1 + 4*k + 4*n)^2), {k, 0, Infinity}], {n, 0, 20}]] (* Vaclav Kotesovec, Nov 14 2017 *)
Numerator[Table[16*Sum[1/(4*k + 1)^2, {k, 0, n - 1} ], {n, 0, 20}]] (* Vaclav Kotesovec, Nov 14 2017 *)

for(n=0,20, print1(numerator(sum(k=0,n-1, 1/(4*k+1)^2)), ", ")) \\ G. C. Greubel, Aug 22 2018

a(n) = numerator of 8*Catalan + Pi^2 - Zeta(2, (4 n + 1)/4), with the Catalan constant given in A006752.

a(n) = numerator(r(n)) with r(n) = Zeta(2, 1/4) - Zeta(2, n + 1/4), with the Hurwitz Zeta function (see the name). With Zeta(2, 1/4) = Psi(1, 1/4) = 8*Catalan + Pi^2 this is the preceding formula, where Psi(1, z) is the Trigamma function. - Wolfdieter Lang, Nov 14 2017

Name simplified and offset set to 0 by Peter Luschny, Nov 14 2017

A173949 a(n) = numerator of (Zeta(2, 1/4) - Zeta(2, n+1/4))/16, where Zeta is the Hurwitz Zeta function.

Original entry on oeis.org

0, 1, 26, 2131, 362164, 105007621, 5156362654, 129102916279, 108696708796264, 13163623138673569, 18033329053484721586, 30330904507928806086691, 30344915637965488890716, 1487479897654682071525709

Offset: 0

Artur Jasinski, Mar 03 2010

For the Catalan constant see A006752.
The denominators are given in A173948.
a(n+1)/A173948(n+1), for n>= 0, gives the partial sum Sum_{k=0..n} 1/(4*k + 1)^2. For {(4*k + 1)^2}A016814.%20The%20limit%20n%20-%3E%20infinity%20is%20given%20in%20A222183%20as%201.074833072...%20.%20-%20_Wolfdieter%20Lang">{k>=0} see A016814. The limit n -> infinity is given in A222183 as 1.074833072... . - _Wolfdieter Lang, Nov 14 2017

			The rationals r(n) begin: 0/1, 1/1, 26/25, 2131/2025, 362164/342225, 105007621/98903025, 5156362654/4846248225, 129102916279/121156205625, 108696708796264/101892368930625, 13163623138673569/12328976640605625, ... - _Wolfdieter Lang_, Nov 14 2017

G. C. Greubel, Table of n, a(n) for n = 0..250
Eric Weisstein's World of Mathematics, Hurwitz Zeta Function
Eric Weisstein's World of Mathematics, Polygamma Function

Cf. A016814, A120268, A173945, A173947, A173948, A222183.

[0] cat [Numerator((&+[1/(4*k+1)^2: k in [0..n-1]])): n in [1..20]]; // G. C. Greubel, Aug 23 2018

r := n -> (Psi(1, 1/4) - Zeta(0, 2, n+1/4))/16:
seq(numer(simplify(r(n))), n=0..13); # Peter Luschny, Nov 14 2017

Table[Numerator[FunctionExpand[(8*Catalan + Pi^2 - Zeta[2, (4*n + 1)/4])/16]], {n, 0, 20}] (* Vaclav Kotesovec, Nov 14 2017 *)
Numerator[Table[Sum[1/(4*k + 1)^2, {k, 0, n-1}], {n, 0, 20}]] (* Vaclav Kotesovec, Nov 14 2017 *)

for(n=0,20, print1(numerator(sum(k=0,n-1, 1/(4*k+1)^2)), ", ")) \\ G. C. Greubel, Aug 23 2018

a(n) = numerator of expression (8*Catalan + Pi^2 - Zeta(2, (4*n + 1)/4))/16.

a(n) = numerator(r(n)) with r(n) = (Zeta(2,1/4) - Zeta(2, n + 1/4))/16, with the Hurwitz Zeta function Z(2, k). With Zeta(2,1/4) = 8 Catalan + Pi^2 this is the preceding formula, and Zeta(2, n + 1/4) = Psi(1, n + 1/4) with the polygamma (trigamma) function Psi(1, k). - Wolfdieter Lang, Nov 14 2017

Edited by Wolfdieter Lang, Nov 14 2017

Name changed according to a formula of Lang by Peter Luschny, Nov 14 2017

A173953 a(n) = numerator of (Zeta(2, 3/4) - Zeta(2, n-1/4)), where Zeta is the Hurwitz Zeta function.

Original entry on oeis.org

0, 16, 928, 119344, 3078464, 1132669904, 606887707616, 49610806397296, 48006150564413056, 48265162121607952, 8192066749392160288, 15200753287254377716912, 33677610844789597790454208

Offset: 1

Artur Jasinski, Mar 03 2010

All numbers in this sequence are divisible by 16. For A173953/16 see A173955.

a(n+2)/A173954(n+2) is, for n >= 0, the partial sum Sum_{k=0..n} 1/(k + 3/4)^2 = 16*Sum_{k=0..n} 1/(4*k + 3)^2. The limit n -> infinity is given in A282824 as Zeta(2, 3/4) = Psi(1, 3/4) = Pi^2 - 8*Catalan, with the trigamma function Psi(1, z) and the Catalan constant A006752.

			The rationals r(n) = Zeta(2, 3/4) - Zeta(2, n-1/4) begin:  0/1, 16/9, 928/441, 119344/53361, 3078464/1334025, 1132669904/481583025, 606887707616/254757420225, 49610806397296/20635351038225, ... - _Wolfdieter Lang_, Nov 14 2017

G. C. Greubel, Table of n, a(n) for n = 1..250
Eric Weisstein's World of Mathematics, Hurwitz Zeta Function
Eric Weisstein's World of Mathematics, Trigamma Function

Denominators are in A173954.

Cf. A006752, A120268, A173945, A173947, A173948, A173949, A173955.

[0] cat [Numerator((&+[16/(4*k+3)^2: k in [0..n-2]])): n in [2..20]]; // G. C. Greubel, Aug 23 2018

r := n -> Zeta(0, 2, 3/4) - Zeta(0, 2, n-1/4):
seq(numer(simplify(r(n))), n=1..13); # Peter Luschny, Nov 14 2017

Table[Numerator[FunctionExpand[Pi^2 - 8*Catalan - Zeta[2, (4*n - 1)/4]]], {n, 1, 20}] (* Vaclav Kotesovec, Nov 14 2017 *)
Numerator[Table[128*n*Sum[(4*k - 1 + 2*n) / ((4*k - 1)^2 * (4*k - 1 + 4*n)^2), {k, 1, Infinity}], {n, 0, 20}]] (* Vaclav Kotesovec, Nov 14 2017 *)
Numerator[Table[16*Sum[1/(4*k + 3)^2, {k, 0, n-1}], {n, 1, 20}]] (* Vaclav Kotesovec, Nov 15 2017 *)

for(n=1,20, print1(numerator(16*sum(k=0,n-2, 1/(4*k+3)^2)), ", ")) \\ G. C. Greubel, Aug 23 2018

a(n) = Numerator of (Pi^2 - 8*Catalan - Zeta(2, (4 n - 1)/4)).
Numerator of 128*n*Sum_{k>=1} (4*k - 1 + 2*n) / ((4*k - 1)^2 * (4*k - 1 + 4*n)^2). - Vaclav Kotesovec, Nov 14 2017
Numerator of 16*Sum_{k=0..n-2} 1/(4*k + 3)^2, n >= 2, with a(1) = 0. See a comment above. - Wolfdieter Lang, Nov 14 2017

Name simplified by Peter Luschny, Nov 14 2017

A173955 a(n) = numerator of (Zeta(2, 3/4) - Zeta(2, n-1/4))/16 where Zeta(n, a) is the Hurwitz Zeta function.

Original entry on oeis.org

0, 1, 58, 7459, 192404, 70791869, 37930481726, 3100675399831, 3000384410275816, 3016572632600497, 512004171837010018, 950047080453398607307, 2104850677799349861903388, 609822785846772474028096357, 611130542819711220012487366

Offset: 1

Artur Jasinski, Mar 03 2010

The denominators are given in A173954.

a(n+2)/A173954(n+2) = (Zeta(2, 3/4) - Zeta(2, n + 7/4))/16 gives, for n >= 0, the partial sum Sum_{k=0..n} 1/(4*n + 3). In the limit n -> infinity the series value is Zeta(2,3/4)/16, with the Hurwitz Zeta function, and it is given in A247037. - Wolfdieter Lang, Nov 15 2017

G. C. Greubel, Table of n, a(n) for n = 1..250

Cf. A006752, A120268, A173945, A173947, A173948, A173949, A173953, A173954, A247037.

[0] cat [Numerator((&+[1/(4*k+3)^2: k in [0..n-2]])): n in [2..20]]; // G. C. Greubel, Aug 23 2018

r := n -> (Zeta(0, 2, 3/4) - Zeta(0, 2, n-1/4))/16:
seq(numer(simplify(r(n))), n=1..15); # Peter Luschny, Nov 14 2017

Table[Numerator[FunctionExpand[(Pi^2 - 8*Catalan - Zeta[2, (4*n - 1)/4])/16]], {n, 1, 20}] (* Vaclav Kotesovec, Nov 14 2017 *)
Numerator[Table[8*n*Sum[(4*k - 1 + 2*n) / ((4*k - 1)^2 * (4*k - 1 + 4*n)^2), {k, 1, Infinity}], {n, 0, 20}]] (* Vaclav Kotesovec, Nov 14 2017 *)
Numerator[Table[Sum[1/(4*k + 3)^2, {k, 0, n-2}], {n, 1, 20}]] (* Vaclav Kotesovec, Nov 15 2017 *)

for(n=1,20, print1(numerator(sum(k=0,n-2, 1/(4*k+3)^2)), ", ")) \\ G. C. Greubel, Aug 23 2018

a(n) = numerator of r(n) with r(n) = (Pi^2 - 8*Catalan - Zeta(2, n - 1/4))/16, with the Hurwitz Zeta function Z(2, z), and the Catalan constant is given in A006752. With Zeta(2, 3/4) = Pi^2 - 8*Catalan this is the formula given in the name.

Numerator of Sum_{k=0..n-2} 1/(4*k + 3)^2, n >= 2, with a(1) = 0. - G. C. Greubel, Aug 23 2018

Numbers changed according to the old (or new) Mathematica program, and edited by Wolfdieter Lang, Nov 14 2017

Name simplified by Peter Luschny, Nov 14 2017

A173954 a(n) = denominator of (Zeta(2, 3/4) - Zeta(2, n-1/4)), where Zeta is the Hurwitz Zeta function.

Original entry on oeis.org

1, 9, 441, 53361, 1334025, 481583025, 254757420225, 20635351038225, 19830572347734225, 19830572347734225, 3351366726767084025, 6196677077792338362225, 13688459664843275442155025

Offset: 1

Artur Jasinski, Mar 03 2010

G. C. Greubel, Table of n, a(n) for n = 1..300

For numerators see A173953.
The Catalan constant is in A006752.
Cf. A006752, A120268, A173945, A173947, A173948, A173949, A173953, A173955.

[1] cat [Denominator((&+[1/(4*k+3)^2: k in [0..n-2]])): n in [2..20]]; // G. C. Greubel, Aug 23 2018

r := n -> Zeta(0, 2, 3/4) - Zeta(0, 2, n-1/4):
seq(denom(simplify(r(n))), n=1..13); # Peter Luschny, Nov 14 2017

Table[Denominator[FunctionExpand[-8*Catalan + Pi^2 - Zeta[2, (4*n - 1)/4]]], {n, 1, 20}] (* Vaclav Kotesovec, Nov 14 2017 *)
Denominator[Table[8*n*Sum[(-1 + 4*k + 2*n) / ((-1 + 4*k)^2*(-1 + 4*k + 4*n)^2), {k, 0, Infinity}], {n, 1, 20}]] (* Vaclav Kotesovec, Nov 14 2017 *)
Denominator[Table[Sum[1/(4*k + 3)^2, {k, 0, n-1}], {n, 1, 20}]] (* G. C. Greubel, Aug 23 2018 *)

for(n=1,20, print1(denominator(sum(k=0,n-2, 1/(4*k+3)^2)), ", ")) \\ G. C. Greubel, Aug 23 2018

a(n) = denominator of (Pi^2 - 8*Catalan - Zeta(2, (4 n - 1)/4)).

a(n) = denominator of Sum_{k=0..(n-2)} 1/(4*k+3)^2. - G. C. Greubel, Aug 23 2018

Name simplified by Peter Luschny, Nov 14 2017

A173982 a(n) = numerator of (Zeta(0,2,1/3) - Zeta(0,2,n+1/3)), where Zeta is the Hurwitz Zeta function.

Original entry on oeis.org

0, 9, 153, 7641, 192789, 32757741, 525987081, 190358321841, 23076404893161, 577743530648769, 578407918658769, 556370890030917009, 160916328686946575601, 220439117509451225357769

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

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

A173982 := proc(n) add( 1/(1/3+i)^2,i=0..n-1) ; numer(%) ; end proc: seq(A173982(n),n=0..20) ; # R. J. Mathar, Apr 22 2010

Table[FunctionExpand[-Zeta[2, (3*n + 1)/3] + Zeta[2, 1/3]], {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(sum(k=0,n-1, 9/(3*k+1)^2)), ", ")) \\ G. C. Greubel, Aug 23 2018

a(n) = numerator 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.
A173982(n)/A173984(n) = sum_{i=0..n} 1/(1/3+i)^2 = 9*sum_{i=0..n} 1/(1+3i)^2 = psi'(1/3) - psi'(n+1/3). - 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

A173983 a(n) = numerator((Zeta(2, 1/3) - Zeta(2, n + 1/3))/9), where Zeta(n, z) is the Hurwitz Zeta function.

Original entry on oeis.org

0, 1, 17, 849, 21421, 3639749, 58443009, 21150924649, 2564044988129, 64193725627641, 64267546517641, 61818987781213001, 17879592076327397289, 24493235278827913928641, 24506988360923903264741

Offset: 0

Artur Jasinski, Mar 04 2010

From Wolfdieter Lang, Nov 12 2017: (Start)
a(n+1)/A173984(n+1) gives, for n >= 0, the partial sum Sum_{k=0..n} 1/(1+3*k)^2.
The limit n -> infinity is given in A214550 as the Hurwitz Zeta function or the Polygamma function (1/9)*Zeta(2, 1/3) = (1/9)*Psi(1, 1/3) = 1.121733...  (End)

			The rationals a(n)/A173984(n) begin 0/1, 1/1, 17/16, 849/784, 21421/19600, 3639749/3312400, 58443009/52998400, 21150924649/19132422400, ... - _Wolfdieter Lang_, Nov 12 2017

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

For denominators see A173984.
For 9*A173983 see A173982.
Cf. A006752, A120268, A173945, A173947, A173948, A173949, A173953, A173955, A173973, A173982-A173987, A214550.

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

a := n -> numer((Zeta(0,2,1/3) - Zeta(0,2,n+1/3))/9):
seq(a(n), n=0..14); # Peter Luschny, Nov 12 2017

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

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

a(n) = numerator of (1/9)(2(Pi^2)/3 + J - Zeta(2,(3n+1)/3)) where J is the constant A173973.

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

Name simplified by Peter Luschny, Nov 12 2017

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

Original entry on oeis.org

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

cp's OEIS Frontend

A003983 Array read by antidiagonals with T(n,k) = min(n,k).

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A173947 a(n) = numerator of (Zeta(2, 1/4) - Zeta(2, n+1/4)), where Zeta is the Hurwitz Zeta function.

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A173949 a(n) = numerator of (Zeta(2, 1/4) - Zeta(2, n+1/4))/16, where Zeta is the Hurwitz Zeta function.

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A173953 a(n) = numerator of (Zeta(2, 3/4) - Zeta(2, n-1/4)), where Zeta is the Hurwitz Zeta function.

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A173955 a(n) = numerator of (Zeta(2, 3/4) - Zeta(2, n-1/4))/16 where Zeta(n, a) is the Hurwitz Zeta function.

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A173954 a(n) = denominator of (Zeta(2, 3/4) - Zeta(2, n-1/4)), where Zeta is the Hurwitz Zeta function.

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