A000464 -id:A000464 - cp's OEIS frontend

A064073 Generalized tangent number d(8,n).

8, 1408, 739328, 806453248, 1506300919808, 4297849713983488, 17390688314209599488, 94727563504456856240128, 668321603392783694711226368, 5928595592752632717848942215168, 64586438563324327821773422563688448, 847680268223550650928681687352090820608

Offset: 1

Eric W. Weisstein, Aug 31 2001

Lars Blomberg, Table of n, a(n) for n = 1..177
D. Shanks, Generalized Euler and class numbers. Math. Comp. 21 (1967) 689-694.
D. Shanks, Corrigenda to: "Generalized Euler and class numbers", Math. Comp. 22 (1968), 699
Eric Weisstein's World of Mathematics, Tangent Number.

Cf. A000464, A064073, A064069, A349267, A349264.

egf := sec(8*x)*2*sin(4*x): ser := series(egf, x, 24):
seq((2*n-1)!*coeff(ser, x, 2*n-1), n = 1..10); # Peter Luschny, Nov 21 2021

E.g.f.: Sum_{k>0} a(k)x^(2k-1)/(2k-1)! = 2*sin(4x)/cos(8x).
a(n) = 2^(4n-1) * A000464(n-1).
a(n) = (2*n-1)!*[x^(2*n-1)](sec(8*x)*2*sin(4*x)). - Peter Luschny, Nov 21 2021

A161722 Generalized Bernoulli numbers B_n(X,0), X a Dirichlet character modulus 8.

Original entry on oeis.org

0, 2, -44, 2166, -196888, 28730410, -6148123332, 1813990148894, -705775346640176, 350112935442888018, -215681051222514096220, 161537815119247080938182, -144555133640020128085896264, 152323571317104251881943249786

Offset: 0

Peter Bala, Jun 18 2009

Let X be a periodic arithmetical function with period m. The generalized Bernoulli polynomials B_n(X,x) attached to X are defined by means of the generating function
(1)... t*exp(t*x)/(exp(m*t)-1) * Sum_{r = 0..m-1} X(r)*exp(r*t) = Sum_{n >= 0} B_n(X,x)*t^n/n!.
The values B_n(X,0) are generalizations of the Bernoulli numbers (case X = 1). For the theory and properties of these polynomials and numbers see [Cohen, Section 9.4]. In the present case, X is chosen to be the Dirichlet character modulus 8 given by
(2)... X(8*n+1) = X(8*n+7) = 1; X(8*n+3) = X(8*n+5) = -1; X(2*n) = 0.
The odd-indexed generalized Bernoulli numbers B_(2*n+1)(X,0) vanish. The current sequence lists the even-indexed values B_(2*n)(X,0).
The coefficients of the generalized Bernoulli polynomials B_n(X,x) are listed in A151751.

H. Cohen, Number Theory - Volume II: Analytic and Modern Tools, Graduate Texts in Mathematics. Springer-Verlag.

Cf. A000464, A153641, A151751, A117442.

G := x*sinh(x)/cosh(2*x): ser := series(G, x, 30):
seq((2*n)!*coeff(ser, x, 2*n), n = 0..14); # Peter Luschny, Nov 26 2020
# After an observation of F. Chapoton in A117442:
A161722 := proc(n) 4^n*add(binomial(2*n, k)*euler(k)*((x+1)/2)^(2*n-k), k=0..2*n);
coeff(%, x, 1) end: seq(A161722(n), n=0..13); # Peter Luschny, Nov 26 2020

terms = 13;
(CoefficientList[x(Sinh[x]/Cosh[2x]) + O[x]^(2terms+3), x] Range[0, 2terms+2]!)[[ ;; ;; 2]] (* Jean-François Alcover, Nov 16 2020 *)

(1)... a(n) = (-1)^(n+1)*2*n*A000464(n-1).
The sequence of generalized Bernoulli numbers
(2)... [B_n(X,0)]n>=2 = [2,0,-44,0,2166,0,...]
has the e.g.f.
(3)... t*(exp(t)-exp(3*t)-exp(5*t)+exp(7*t))/(exp(8*t)-1),
which simplifies to
(4)... t*sinh(t)/cosh(2*t) = 2*t^2/2! - 44*t^4/4! + ....
Hence
(5)... B_(2*n)(X,0) = (-1)^(n+1)*2*n*A000464(n-1) and B_(2*n+1)(X,0) = 0.
a(n) = (-1/2)*16^n*n*euler(2*n-1, 1/4) for n >= 1 after a formula of Peter Bala in A000464. - Peter Luschny, Nov 26 2020

Cross-reference corrected by Peter Bala, Jun 22 2009

Offset set to 0 and a(0) = 0 prepended by Peter Luschny, Nov 26 2020

A000822 Expansion of (sin^2 x + sin x) /cos 2x.

Original entry on oeis.org

0, 1, 2, 11, 40, 361, 1952, 24611, 177280, 2873041, 25866752, 512343611, 5535262720, 129570724921, 1633165156352, 44110959165011, 635421069967360, 19450718635716001, 315212388819402752, 10784052561125704811

Offset: 0

N. J. A. Sloane

nonn

R. J. Mathar, Table of n, a(n) for n = 0..177

Bisections are A000464 and A000816.

A098432 Coefficients of polynomials S(n,x) related to Springer numbers.

Original entry on oeis.org

1, 8, 7, 128, 304, 177, 3072, 13952, 21080, 10199, 98304, 724992, 2016000, 2441056, 1051745, 3932160, 42762240, 187643904, 407505664, 428605352, 169913511, 188743680, 2839019520, 17974591488, 60428242944, 111985428352

Offset: 0

Ralf Stephan, Sep 07 2004

			S(0,x) = 1,
S(1,x) = 8*x + 7,
S(2,x) = 128*x^2 + 304*x + 177,
S(3,x) = 3072*x^3 + 13952*x^2 + 21080*x + 10199.

A. Randrianarivony and J. Zeng, Une famille des polynomes qui interpole plusieurs suites..., Adv. Appl. Math. 17 (1996), 1-26.

Cf. A001586. S(n, 1/2) = A000464(n+1), S(n, -1/2) = A000281(n).
Leading coefficients are A051189. Constant terms are in A098433.
Cf. A001586. S(n, 1/2) = A000464(n), S(n, -1/2) = A000281(n).

S(n,x)=if(n<1,1,(2*x+2)*(2*x+4)*S(n-1,x+2)-(2*x+1)^2*S(n-1,x))

Recurrence: S(0, x)=1, S(n, x)=(2x+2)(2x+4)S(n-1, x+2)-(2x+1)^2S(n-1, x).

G.f.: Sum[n>=0, S(n, x)t^n] = 1/(1+t-4*2(x+1)t/(1-4*2(x+2)t/(1+t-4*4(x+3)t/(1-4+4(x+4)t/...)))).

A101923 Expansion of 2 * arccot(cos(x)).

Original entry on oeis.org

1, 2, 1, -148, -3719, -20098, 5055961, 403644152, 7831409041, -2707151879398, -472143935754479, -22085804322342748, 9362259685093715401, 2995219209329323622102, 274269338931958691728681, -132963342779629343323496848, -70698673853383423350187244639

Offset: 1

Ralf Stephan, Dec 27 2004

Odd coefficients are zero.

Vincenzo Librandi, Table of n, a(n) for n = 1..250

Cf. A012494, A000364, A000464, A156138, A002439.

Cf. other sequences with a g.f. of the form sin(x)/(1 - k*sin^2(x)): A012494 (k=-1), A000364 (k=1), A000464 (k=2), A156138 (k=3), A002439 (k=4).

with(gfun):
series(sin(x)/(1-(1/2)*sin(x)^2), x, 35):
L := seriestolist(%):
seq(op(2*i, L)*(2*i-1)!, i = 1..floor((1/2)*nops(L)));
# Peter Bala, Feb 06 2017

With[{nn=40},Take[CoefficientList[Series[2ArcCot[Cos[x]],{x,0,nn}],x] Range[0,nn]!,{3,-1,2}]] (* Harvey P. Dale, Nov 17 2014 *) (* adapted by Vincenzo Librandi, Feb 07 2017 *)

2*acot(cos(x)) = Pi/2 + x^2/2! + 2*x^4/4! + x^6/6! - 148*x^8/8! - 3719*x^10/10! -...
2*atan(cos(x)) = Pi/2 - x^2/2! - 2*x^4/4! - x^6/6! + 148*x^8/8! + 3719*x^10/10! +...
G.f. sin(x)/(1 - 1/2*sin(x)^2) = x + 2*x^3/3! + x^5/5! - 148*x^7/7! - ... - Peter Bala, Feb 06 2017

More terms from Harvey P. Dale, Nov 17 2014

Signs of the data entries corrected by Peter Bala, Feb 06 2017

A151751 Triangle of coefficients of generalized Bernoulli polynomials associated with a Dirichlet character modulus 8.

Original entry on oeis.org

2, 0, 6, -44, 0, 12, 0, -220, 0, 20, 2166, 0, -660, 0, 30, 0, 15162, 0, -1540, 0, 42, -196888, 0, 60648, 0, -3080, 0, 56, 0, -1771992, 0, 181944, 0, -5544, 0, 72, 28730410, 0, -8859960, 0, 454860, 0, -9240, 0, 90

Offset: 2

Peter Bala, Jun 17 2009

Let X be a periodic arithmetical function with period m. The generalized Bernoulli polynomials B_n(X,x) attached to X are defined by means of the generating function
(1)... t*exp(t*x)/(exp(m*t)-1) * sum {r = 0..m-1} X(r)*exp(r*t)
= sum {n = 0..inf} B_n(X,x)*t^n/n!.
For the theory and properties of these polynomials see [Cohen, Section 9.4]. In the present case, X is chosen to be the Dirichlet character modulus 8 given by
(2)... X(8*n+1) = X(8*n+7) = 1; X(8*n+3) = X(8*n+5) = -1; X(2*n) = 0.
Cf. A153641.

			The triangle begins
n\k|........0.......1........2.......3......4.......5.......6
=============================================================
.2.|........2
.3.|........0.......6
.4.|......-44.......0.......12
.5.|........0....-220........0......20
.6.|.....2166.......0.....-660.......0......30
.7.|........0...15162........0...-1540.......0.....42
.8.|..-196888.......0....60648.......0...-3080......0......56
...

H. Cohen, Number Theory - Volume II: Analytic and Modern Tools, Graduate Texts in Mathematics. Springer-Verlag.

Cf. A000111, A000464, A000828, A001586, A109572, A153641, A161722.

with(gfun):
for n from 2 to 10 do
Genbernoulli(n,x) := 8^(n-1)*(bernoulli(n,(x+1)/8)-bernoulli(n,(x+3)/8)-bernoulli(n,(x+5)/8)+bernoulli(n,(x+7)/8));
seriestolist(series(Genbernoulli(n,x),x,10))
end do;

TABLE ENTRIES
(1)... T(2*n,2*k+1) = 0, T(2*n+1,2*k) = 0;
(2)... T(2*n,2*k) = (-1)^(n-k-1)*C(2*n,2*k)*2*(n-k)*A000464(n-k-1);
(3)... T(2*n+1,2*k+1) = (-1)^(n-k-1)*C(2*n+1,2*k+1)*2*(n-k)*A000464(n-k-1);
where C(n,k) = binomial(n,k).
GENERATING FUNCTION
The e.g.f. for these generalized Bernoulli polynomials is
(4)... t*exp(x*t)*(exp(t)-exp(3*t)-exp(5*t)+exp(7*t))/(exp(8*t)-1)
= sum {n = 2..inf} B_n(X,x)*t^n/n! = 2*t^2/2! + 6*x*t^3/3! + (12*x^2 - 44)*t^4/4! + ....
In terms of the ordinary Bernoulli polynomials B_n(x)
(5)... B_n(X,x) = 8^(n-1)*{B_n((x+1)/8) - B_n((x+3)/8) - B_n((x+5)/8) + B_n((x+7)/8)}.
The B_n(X,x) are Appell polynomials of the form
(6)... B_n(X,x) = sum {j = 0..n} binomial(n,j)*B_j(X,0)*x*(n-j).
The sequence of generalized Bernoulli numbers
(7)... [B_n(X,0)]n>=2 = [2,0,-44,0,2166,0,...]
has the e.g.f.
(8)... t*(exp(t)-exp(3*t)-exp(5*t)+exp(7*t))/(exp(8*t)-1),
which simplifies to
(9)... t*sinh(t)/cosh(2*t).
Hence
(10)... B_(2*n)(X,0) = (-1)^(n+1)*2*n*A000464(n-1); B_(2*n+1)(X,0) = 0.
The sequence {B_(2*n)(X,0)}n>=2 is A161722.
RELATION WITH TWISTED SUMS OF POWERS
The generalized Bernoulli polynomials may be used to evaluate sums of k-th powers twisted by the function X(n). For the present case the result is
(11)... sum{n = 0..8*N-1} X(n)*n^k = 1^k-3^k-5^k+7^k- ... +(8*N-1)^k
= [B_(k+1)(X,8*N) - B_(k+1)(X,0)]/(k+1)
For the proof, apply [Cohen, Corollary 9.4.17 with m = 8 and x = 0].
MISCELLANEOUS
(12)... Row sums [2, 6, -32, ...] = (-1)^(1+binomial(n,2))*A109572(n)
= (-1)^(1+binomial(n,2))*n*A000828(n-1) = (-1)^(1+binomial(n,2))*n* 2^(n-2)*A000111(n-1).

A156138 Q_{2n+1}(sqrt(2))/sqrt(2) (see A104035).

Original entry on oeis.org

1, 17, 901, 99917, 18991081, 5514615017, 2270974911661, 1258937450889317, 903952433274722641, 816101554527859690817, 904827968753139590344021, 1208617989532834039606507517, 1914312457105234828011498655801, 3547500444096776665586928259547417, 7604155838367549221056955383942297981

Offset: 0

N. J. A. Sloane, Nov 06 2009

			G.f. = 1 + 17*x + 901*x^2 + 99917*x^3 + 18991081*x^4 + 5514615017*x^5 + ... - _Michael Somos_, Aug 19 2018

G. C. Greubel, Table of n, a(n) for n = 0..207

Cf. A101923, A000364, A000464, A002439.

Cf. other sequences with a g.f. of the form sin(x)/(1 - k*sin^2(x)): A101923 (k=1/2), A000364 (k=1), A000464 (k=2), A002439 (k=4).

with(gfun):
series(sin(x)/(1-3*sin(x)^2), x, 30):
L := seriestolist(%):
seq(op(2*i, L)*(2*i-1)!, i = 1..floor((1/2)*nops(L)));
# Peter Bala, Feb 06 2017

With[{nmax = 50}, CoefficientList[Series[Sin[x]/(1 - 3*Sin[x]^2), {x, 0, nmax}], x]*Range[0, nmax]!][[2 ;; ;; 2]] (* G. C. Greubel, Aug 17 2018 *)

x='x+O('x^50); v=Vec(serlaplace(sin(x)/(1 - 3*sin(x)^2))); vector((#v-1)\2 ,n,v[2*n-1]) \\ G. C. Greubel, Aug 17 2018

E.g.f.: sin(x)/(1 - 3*sin(x)^2) = x + 17*x^3/3! + 901*x^5/5! + 99917*x^7/7! + ... - Peter Bala, Feb 06 2017

A262144 Square array read by antidiagonals upwards: the n-th row o.g.f. is exp( Sum_{i >= 1} d(n,i+1)*x^i/i ) for n >= 1, where d(n,k) is Shanks's array of generalized Euler and class numbers.

Original entry on oeis.org

1, 1, 2, 1, 11, 10, 1, 46, 241, 108, 1, 128, 2739, 10411, 2214, 1, 272, 16384, 265244, 836321, 75708, 1, 522, 64964, 2883584, 45094565, 112567243, 3895236, 1, 904, 212325, 18852096, 822083584, 12975204810, 22949214033

Offset: 1

Peter Bala, Sep 18 2015

Shanks's array d(n,k) n >= 1, k >= 1, is A235606.
We conjecture that the entries of the present array are all integers. More generally, we conjecture that for r = 1, 2, ... and for each n >= 1, the expansion of exp( Sum_{i >= 1} d(n,i + r)*x^i/i ) has integer coefficients. This is the case r = 1.
For the similarly defined array associated with Shanks' c(n,k) array see A262143.

			The triangular array begins
1
1   2
1  11     10
1  46    241      108
1 128   2739    10411      2214
1 272  16384   265244    836321       75708
1 522  64964  2883584  45094565   112567243     3895236
1 904 212325 18852096 822083584 12975204810 22949214033 ...
The square array begins (row indexing n starts at 1)
1, 2, 10, 108, 2214, 75708, 3895236, 280356120, 26824493574, ...
1, 11, 241, 10411, 836321, 112567243, 22949214033, 6571897714923, 2507281057330113, ...
1, 46, 2739, 265244, 45094565, 12975204810, 5772785327575, 3656385436507960, 3107332328608143945, ...
1, 128, 16384, 2883584, 822083584, 395136991232, 300338473074688, 330739694704787456, 493338658405976375296, ...
1, 272, 64864, 18852096, 8133183744, 5766226378752, 6562478680375296, 11019751545852395520, 25333348417380699340800, ...
1, 522, 212325, 94501768, 57064909374, 54459242196516, 84430282319806062, 197625548666434041000, 642556291067409622713543, ...
1, 904, 586452, 382674008, 311514279098, 379982635729752, 753288329161251844, 2308779464340711480136, 10003494921382094286802995, ...

P. Bala, Notes on logarithmic differentiation, the binomial transform and series reversion
William Y. C. Chen, Neil J. Y. Fan, Jeffrey Y. T. Jia , The generating function for the Dirichlet series Lm(s) Mathematics of Computation, Vol. 81, No. 278, April 2012.
D. Shanks, Generalized Euler and class numbers. Math. Comp. 21 (1967) 689-694.
D. Shanks, Corrigenda to: "Generalized Euler and class numbers", Math. Comp. 22 (1968), 699.
D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 689-694; 22 (1968), 699. [Annotated scanned copy]

Cf. A000182 (d(1,n)), A000464 (d(2,n)), A000191 (d(3,n)), A000318 (d(4,n)), A000320 (d(5,n)), A000411 (d(6,n)), A064072 (d(7,n)), A235605, A235606, A262143, A262145 (row 1 of square array).

A370411 Square array T(n, k) = denominator( zeta_r(2n) sqrt(A003658(k + 2)) / Pi^(4*n) ), read by antidiagonals, where zeta_r is the Dedekind zeta-function over r and r is the real quadratic field with discriminant A003658(k + 2).

Original entry on oeis.org

1, 75, 1, 16875, 24, 1, 221484375, 34560, 18, 1, 116279296875, 116121600, 58320, 39, 1, 12950606689453125, 780337152000, 440899200, 296595, 51, 1, 4861333986053466796875, 8899589151129600, 6666395904000, 68420017575, 663255, 63, 1, 677114376628875732421875

Offset: 0

Thomas Scheuerle, Feb 22 2024

			The array begins:
           1,            1,             1,              1,                 1
          75,           24,            18,             39,                51
       16875,        34560,         58320,         296595,            663255
   221484375,    116121600,     440899200,    68420017575,       20126472975
116279296875, 780337152000, 6666395904000, 93393323989875, 10382542981248375

Index entries for zeta function.

Cf. A370412 (numerators).
Cf. A003658, A080891, A097916, A097917, A230375, A370413.
Cf. A000191, A000411, A000464, A064072, A064075.
Cf. A002432 (denominators zeta(2*n)/Pi^(2*n)).
Cf. A046988 (numerators zeta(2*n)/Pi^(2*n)).
Coefficients of Dedekind zeta functions for real quadratic number fields of discriminants 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, respectively.

\p 700
row(n) = {v=[]; for(k=2, 30, if(isfundamental(k), v=concat(v, denominator(bestappr(sqrt(k)*lfun(x^2-(k%2)*x-floor(k/4), 2*n)/Pi^(4*n)))))); v}
z(n,d) = if(n == 0, 0,(1/(-2*n))*bernfrac(2*n)*d^(2*n-1)*sum(k=1,d-1, kronecker(d, k)*subst(bernpol(2*n),x,k/d)*(1/(-2*n))))
row(n) = {v=[]; for(k=2, 100, if(isfundamental(k), v=concat(v, denominator((2^(n*4)*n^2*z(n,k))/((2*n)!^2 * (k^(2*n-1))))))); v} \\ more accuracy here

# Only suitable for small n and k
def T(n, k):
    discs = [fundamental_discriminant(i) for i in range(1, 4*k+10)]
    D = sorted(list(set(discs)))[k+1]
    zetaK = QuadraticField(D).zeta_function(1000)
    val = (zetaK(2*n)*sqrt(D)/(pi^(4*n))).n(1000).nearby_rational(2^-900)
    return val.denominator() # Robin Visser, Mar 19 2024

T(n, k) = denominator( 2^(n*4) * n^2 * zeta_r(1 - 2*n) /((2*n)!^2 * A003658(k + 2)^(2*n - 1)) ), where zeta_r is the Dedekind zeta-function over r and r is the real quadratic field with discriminant A003658(k + 2).
T(n, 0) = denominator((5^(-2*n)*(zeta(2*n, 1/5) - zeta(2*n, 2/5) - zeta(2*n, 3/5) + zeta(2*n, 4/5) ))*zeta(2*n)*sqrt(5)*Pi^(-4*n)). A sum of Hurwitz zeta functions with signs according A080891.
T(n, 1) = denominator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * A000464(n+1) /((2*n)!^2 * 8^(2*n - 1)) ).
T(n, 2) = denominator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * A000191(n+1) /((2*n)!^2 * 12^(2*n - 1)) ).
T(n, 3) = denominator((13^(-2*n)*(zeta(2*n, 1/13) - zeta(2*n, 2/13) + zeta(2*n, 3/13) + zeta(2*n, 4/13) - zeta(2*n, 5/13) - zeta(2*n, 6/13) - zeta(2*n, 7/13) -  zeta(2*n, 8/13) + zeta(2*n, 9/13) + zeta(2*n, 10/13) - zeta(2*n, 11/13) + zeta(2*n, 12/13) ))*zeta(2*n)*sqrt(13)*Pi^(-4*n)). A sum of Hurwitz zeta functions with signs according the Dirichlet character X13(12,.).
T(n, 6) = denominator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * A000411(n+1) /((2*n)!^2 * 24^(2*n - 1)) ).
T(n, 7) = denominator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * A064072(n+1) /((2*n)!^2 * 28^(2*n - 1)) ).
T(n, 11) = denominator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * A064075(n+1) /((2*n)!^2 * 40^(2*n - 1)) ).
T(n, k) = denominator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * d(A003658(k+2)/4, n+1) /((2*n)!^2 * 40^(2*n - 1)) ), for all k where A003658(k+2) is a multiple of four (The discriminant of the quadratic field is from 4*A230375). d() are the generalized tangent numbers.
T(0, k) = 1, because for a real quadratic number field the discriminant D is positive, hence the Kronecker symbol (D/-1) = 1. This means the associated Dirichlet L-function will be zero at s = 0 inside the expression zeta_r(s) = zeta(s)*L(s, x).

A370412 Square array T(n, k) = numerator( zeta_r(2n) sqrt(A003658(k + 2)) / Pi^(4*n) ), read by antidiagonals, where zeta_r is the Dedekind zeta-function over r and r is the real quadratic field with discriminant A003658(k + 2).

Original entry on oeis.org

0, 2, 0, 4, 1, 0, 536, 11, 1, 0, 2888, 361, 23, 2, 0, 3302008, 24611, 1681, 116, 4, 0, 12724582576, 2873041, 257543, 267704, 328, 4, 0, 18194938976, 27233033477, 67637281, 3741352, 92656, 88, 1, 0, 875222833138832, 11779156811, 18752521534133, 1156377368, 479214352, 287536, 29, 2, 0

Offset: 0

Thomas Scheuerle, Feb 22 2024

			The array begins:
          0,           0,              0,               0,                 0
          2,           1,              1,               2,                 4
          4,          11,             23,             116,               328
        536,         361,           1681,          267704,             92656
       2888,       24611,         257543,         3741352,         479214352
    3302008,     2873041,       67637281,      1156377368,       14816172016
12724582576, 27233033477, 18752521534133, 753075777246704, 16476431095568992

Index entries for zeta function.

Cf. A370411 (denominators).
Cf. A003658, A080891, A097916, A097917, A230375, A370413.
Cf. A000191, A000411, A000464, A064072, A064075.
Cf. A002432 (denominators zeta(2*n)/Pi^(2*n)).
Cf. A046988 (numerators zeta(2*n)/Pi^(2*n)).
Coefficients of Dedekind zeta functions for real quadratic number fields of discriminants 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, respectively.

\p 700
row(n) = {v=[]; for(k=2, 50, if(isfundamental(k), v=concat(v, numerator(bestappr(sqrt(k)*lfun(x^2-(k%2)*x-floor(k/4), 2*n)/Pi^(4*n)))))); v}
z(n,d) = if(n == 0, 0,(1/(-2*n))*bernfrac(2*n)*d^(2*n-1)*sum(k=1,d-1, kronecker(d, k)*subst(bernpol(2*n),x,k/d)*(1/(-2*n))))
row(n) = {v=[]; for(k=2, 100, if(isfundamental(k), v=concat(v, numerator((2^(n*4)*n^2*z(n,k))/((2*n)!^2 * (k^(2*n-1))))))); v} \\ more accuracy here

# Only suitable for small n and k
def T(n, k):
    discs = [fundamental_discriminant(i) for i in range(1, 4*k+10)]
    D = sorted(list(set(discs)))[k+1]
    zetaK = QuadraticField(D).zeta_function(1000)
    val = (zetaK(2*n)*sqrt(D)/(pi^(4*n))).n(1000).nearby_rational(2^-900)
    return val.numerator() # Robin Visser, Mar 19 2024

T(n, k) = numerator( 2^(n*4) * n^2 * zeta_r(1 - 2*n) /((2*n)!^2 * A003658(k + 2)^(2*n - 1)) ), where zeta_r is the Dedekind zeta-function over r and r is the real quadratic field with discriminant A003658(k + 2).
T(n, 0) = numerator((5^(-2*n)*(zeta(2*n, 1/5) - zeta(2*n, 2/5) - zeta(2*n, 3/5) + zeta(2*n, 4/5) ))*zeta(2*n)*sqrt(5)*Pi^(-4*n)). A sum of Hurwitz zeta functions with signs according A080891.
T(n, 1) = numerator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * A000464(n+1) /((2*n)!^2 * 8^(2*n - 1)) ).
T(n, 2) = numerator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * A000191(n+1) /((2*n)!^2 * 12^(2*n - 1)) ).
T(n, 3) = numerator((13^(-2*n)*(zeta(2*n, 1/13) - zeta(2*n, 2/13) + zeta(2*n, 3/13) + zeta(2*n, 4/13) - zeta(2*n, 5/13) - zeta(2*n, 6/13) - zeta(2*n, 7/13) -  zeta(2*n, 8/13) + zeta(2*n, 9/13) + zeta(2*n, 10/13) - zeta(2*n, 11/13) + zeta(2*n, 12/13) ))*zeta(2*n)*sqrt(13)*Pi^(-4*n)). A sum of Hurwitz zeta functions with signs according the Dirichlet character X13(12,.).
T(n, 6) = numerator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * A000411(n+1) /((2*n)!^2 * 24^(2*n - 1)) ).
T(n, 7) = numerator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * A064072(n+1) /((2*n)!^2 * 28^(2*n - 1)) ).
T(n, 11) = numerator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * A064075(n+1) /((2*n)!^2 * 40^(2*n - 1)) ).
T(n, k) = numerator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * d(A003658(k+2)/4, n+1) /((2*n)!^2 * 40^(2*n - 1)) ), for all k where A003658(k+2) is a multiple of four (The discriminant of the quadratic field is from 4*A230375). d() are the generalized tangent numbers.
T(0, k) = 0, because for a real quadratic number field the discriminant D is positive, hence the Kronecker symbol (D/-1) = 1. This means the associated Dirichlet L-function will be zero at s = 0 inside the expression zeta_r(s) = zeta(s)*L(s, x).

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A370411 Square array T(n, k) = denominator( zeta_r(2n) sqrt(A003658(k + 2)) / Pi^(4*n) ), read by antidiagonals, where zeta_r is the Dedekind zeta-function over r and r is the real quadratic field with discriminant A003658(k + 2).

A370412 Square array T(n, k) = numerator( zeta_r(2n) sqrt(A003658(k + 2)) / Pi^(4*n) ), read by antidiagonals, where zeta_r is the Dedekind zeta-function over r and r is the real quadratic field with discriminant A003658(k + 2).