cp's OEIS Frontend

A370411 Square array T(n, k) = denominator( zeta_r(2*n) * 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).

%I A370411 #42 Apr 30 2024 08:28:11
%S A370411 1,75,1,16875,24,1,221484375,34560,18,1,116279296875,116121600,58320,
%T A370411 39,1,12950606689453125,780337152000,440899200,296595,51,1,
%U A370411 4861333986053466796875,8899589151129600,6666395904000,68420017575,663255,63,1,677114376628875732421875
%N A370411 Square array T(n, k) = denominator( zeta_r(2*n) * 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).
%H A370411 <a href="/index/Z#zeta_function">Index entries for zeta function</a>.
%F A370411 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).
%F A370411 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.
%F A370411 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)) ).
%F A370411 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)) ).
%F A370411 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,.).
%F A370411 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)) ).
%F A370411 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)) ).
%F A370411 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)) ).
%F A370411 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.
%F A370411 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).
%e A370411 The array begins:
%e A370411            1,            1,             1,              1,                 1
%e A370411           75,           24,            18,             39,                51
%e A370411        16875,        34560,         58320,         296595,            663255
%e A370411    221484375,    116121600,     440899200,    68420017575,       20126472975
%e A370411 116279296875, 780337152000, 6666395904000, 93393323989875, 10382542981248375
%o A370411 (PARI)
%o A370411 \p 700
%o A370411 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}
%o A370411 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))))
%o A370411 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
%o A370411 (Sage)  # Only suitable for small n and k
%o A370411 def T(n, k):
%o A370411     discs = [fundamental_discriminant(i) for i in range(1, 4*k+10)]
%o A370411     D = sorted(list(set(discs)))[k+1]
%o A370411     zetaK = QuadraticField(D).zeta_function(1000)
%o A370411     val = (zetaK(2*n)*sqrt(D)/(pi^(4*n))).n(1000).nearby_rational(2^-900)
%o A370411     return val.denominator() # _Robin Visser_, Mar 19 2024
%Y A370411 Cf. A370412 (numerators).
%Y A370411 Cf. A003658, A080891, A097916, A097917, A230375, A370413.
%Y A370411 Cf. A000191, A000411, A000464, A064072, A064075.
%Y A370411 Cf. A002432 (denominators zeta(2*n)/Pi^(2*n)).
%Y A370411 Cf. A046988 (numerators zeta(2*n)/Pi^(2*n)).
%Y A370411 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.
%K A370411 nonn,tabl,frac
%O A370411 0,2
%A A370411 _Thomas Scheuerle_, Feb 22 2024