A035187 -id:A035187 - cp's OEIS frontend

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).

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).

A031366 Number of symmetrically inequivalent coincidence rotations of icosian ring of index n.

Original entry on oeis.org

1, 0, 0, 25, 36, 0, 0, 0, 100, 0, 288, 0, 0, 0, 0, 440, 0, 0, 800, 900, 0, 0, 0, 0, 960, 0, 0, 0, 1800, 0, 2048, 0, 0, 0, 0, 2500, 0, 0, 0, 0, 3528, 0, 0, 7200, 3600, 0, 0, 0, 2550, 0, 0, 0, 0, 0, 10368, 0, 0, 0, 7200, 0, 7688, 0, 0, 7330, 0, 0, 0, 0, 0, 0, 10368, 0, 0, 0, 0, 20000, 0, 0, 12800, 15840, 8362, 0, 0, 0, 0, 0, 0, 0, 16200, 0, 0, 0, 0, 0, 28800, 0, 0, 0, 28800, 23899

Offset: 1

N. J. A. Sloane

The overall number of coincidence rotations is 7200 times this value. Some symmetrically distinct rotations generate the same coincidence site modules, hence a(n) >= A331143(n). - Andrey Zabolotskiy, Feb 16 2021

M. Baake, Solution of the coincidence problem in dimensions d <= 4, in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.; arXiv:math/0605222 [math.MG], 2006.
Michael Baake and Peter Zeiner, Coincidences in 4 dimensions, Phil. Mag. 88 (2008), 2025-2032; arXiv:0712.0363 [math.MG]. See Section 4. Caution: there is a typo in a(19).

Cf. A331143.

read("transforms") :
# expansion of 1/(1-5^(-s)) in (5.10)
L1 := [1,seq(0,i=2..200)] :
for k from 1 do
    if 5^k <= nops(L1) then
        L1 := subsop(5^k=1,L1) ;
    else
        break ;
    end if;
end do:
# multiplication with 1/(1-p^(-2s)) in (5.10)
for i from 1 do
    p := ithprime(i) ;
    if modp(p,5) = 2 or modp(p,5)=3 then
        Laux := [1,seq(0,i=2..200)] :
        for k from 1 do
            if p^(2*k) <= nops(Laux) then
                Laux := subsop(p^(2*k)=1,Laux) ;
            else
                break ;
            end if;
        end do:
        L1 := DIRICHLET(L1,Laux) ;
    end if;
    if p > nops(L1) then
        break;
    end if;
end do:
# multiplication with 1/(1-p^(-s))^2 in (5.10)
for i from 1 do
    p := ithprime(i) ;
    if modp(p,5) = 1 or modp(p,5)=4 then
        Laux := [1,seq(0,i=2..200)] :
        for k from 1 do
            if p^k <= nops(Laux) then
                Laux := subsop(p^k=k+1,Laux) ;
            else
                break ;
            end if;
        end do:
        L1 := DIRICHLET(L1,Laux) ;
    end if;
    if p > nops(L1) then
        break;
    end if;
end do:
# this is now Zeta_L(s), seems to be A035187
# print(L1) ;
# generate Zeta_L(s-1)
L1shft := [seq(op(i,L1)*i,i=1..nops(L1))] ;
# generate 1/Zeta_L(s)
L1x := add(op(i,L1)*x^(i-1),i=1..nops(L1)) :
taylor(1/L1x,x=0,nops(L1)) :
L1i := gfun[seriestolist](%) ;
# generate 1/Zeta_L(2s)
L1i2 := [1,seq(0,i=2..nops(L1))] ;
for k from 2 to nops(L1i) do
    if k^2 < nops(L1i2) then
        L1i2 := subsop(k^2=op(k,L1i),L1i2) ;
    else
        break ;
    end if;
end do:
# generate Zeta_L(s)*Zeta_L(s-1)
DIRICHLET(L1,L1shft) ;
# generate Zeta_L(s)*Zeta_L(s-1)/Zeta_L(2s) = Phi(s)
Phis := DIRICHLET(%,L1i2) ;
# generate Phis(s-1)
Phishif := [seq(op(i,Phis)*i,i=1..nops(Phis))] ;
DIRICHLET(Phis,Phishif) ;

See Baake (1997) for the Dirichlet g.f.

Terms beyond a(36) from R. J. Mathar, Mar 04 2018

New name from Andrey Zabolotskiy, Feb 16 2021

A300382 Dirichlet series for a cubic module of rank 6.

Original entry on oeis.org

1, 0, 0, 8, 6, 0, 0, 0, 10, 0, 24, 0, 0, 0, 0, 32, 0, 0, 40, 48, 0, 0, 0, 0, 30, 0, 0, 0, 60, 0, 64, 0, 0, 0, 0, 80, 0, 0, 0, 0, 84, 0, 0, 192, 60, 0, 0, 0, 51, 0, 0, 0, 0, 0, 144, 0, 0, 0, 120, 0, 124, 0, 0, 130, 0, 0, 0, 0, 0, 0, 144, 0, 0, 0, 0, 320, 0, 0, 160, 192, 91, 0, 0, 0, 0, 0, 0, 0, 180, 0, 0, 0, 0, 0, 240, 0, 0, 0, 240, 239, 204, 0, 0, 0, 0, 0, 0, 0, 220, 0, 0, 0, 0, 0, 0, 480, 0, 0, 0, 0, 405

Offset: 1

R. J. Mathar, Mar 04 2018

Submitted as a substitute for A031365 which appears to display a faulty A031365(16)=24 in the version published 1997.

M. Baake, Solution of the coincidence problem in dimensions d<=4, arxiv:math/0605222 (2006), (5.12)

Cf. A031365.

read("transforms") :
# expansion of 1/(1-5^(-s)) in (5.10)
L1 := [1,seq(0,i=2..200)] :
for k from 1 do
    if 5^k <= nops(L1) then
        L1 := subsop(5^k=1,L1) ;
    else
        break ;
    end if;
end do:
# multiplication with 1/(1-p^(-2s)) in (5.10)
for i from 1 do
    p := ithprime(i) ;
    if modp(p,5) = 2 or modp(p,5)=3 then
        Laux := [1,seq(0,i=2..200)] :
        for k from 1 do
            if p^(2*k) <= nops(Laux) then
                Laux := subsop(p^(2*k)=1,Laux) ;
            else
                break ;
            end if;
        end do:
        L1 := DIRICHLET(L1,Laux) ;
    end if;
    if p > nops(L1) then
        break;
    end if;
end do:
# multiplication with 1/(1-p^(-s))^2 in (5.10)
for i from 1 do
    p := ithprime(i) ;
    if modp(p,5) = 1 or modp(p,5)=4 then
        Laux := [1,seq(0,i=2..200)] :
        for k from 1 do
            if p^k <= nops(Laux) then
                Laux := subsop(p^k=k+1,Laux) ;
            else
                break ;
            end if;
        end do:
        L1 := DIRICHLET(L1,Laux) ;
    end if;
    if p > nops(L1) then
        break;
    end if;
end do:
# this is now Zeta_L(s), seems to be A035187
# print(L1) ;
# generate Zeta_L(s-1)
L1shft := [seq(op(i,L1)*i,i=1..nops(L1))] ;
# generate 1/Zeta_L(s)
L1x := add(op(i,L1)*x^(i-1),i=1..nops(L1)) :
taylor(1/L1x,x=0,nops(L1)) :
L1i := gfun[seriestolist](%) ;
# generate 1/Zeta_L(2s)
L1i2 := [1,seq(0,i=2..nops(L1))] ;
for k from 2 to nops(L1i) do
    if k^2 < nops(L1i2) then
        L1i2 := subsop(k^2=op(k,L1i),L1i2) ;
    else
        break ;
    end if;
end do:
# generate Zeta_L(s)*Zeta_L(s-1)
DIRICHLET(L1,L1shft) ;
# generate Zeta_L(s)*Zeta_L(s-1)/Zeta_L(2s)
L1 := DIRICHLET(%,L1i2) ;
# generate 1/(1+4^(-s))
Laux := [1,seq(0,i=2..nops(L1))] :
for k from 1 do
    if 4^k <= nops(Laux) then
        Laux := subsop(4^k=(-1)^k,Laux) ;
    else
        break;
    end if ;
end do:
# generate Zeta_L(s)*Zeta_L(s-1)/Zeta_L(2s)/(1+4^(-s))
L1 := DIRICHLET(L1,Laux) ;
# generate 1+4^(1-s)
Laux := [1,seq(0,i=2..3),4,seq(0,i=5..nops(L1))] ;
DIRICHLET(L1,Laux) ; # R. J. Mathar, Mar 04 2018

A327785 Square array read by antidiagonals: A(n,k) = Sum_{d|n} (k/d), (n>=1, k>=0), where (m/n) is the Kronecker symbol.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 0, 0, 3, 1, 1, 1, 1, 1, 2, 1, 1, 0, 2, 1, 0, 4, 1, 1, 1, 0, 1, 0, 0, 2, 1, 1, 2, 1, 1, 2, 0, 2, 4, 1, 1, 1, 2, 1, 1, 2, 0, 1, 3, 1, 1, 2, 0, 3, 2, 0, 2, 0, 1, 4, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 2, 3, 0, 4, 0, 0, 3, 0, 0, 6, 1

Offset: 1

Seiichi Manyama, Sep 25 2019

			Square array begins:
   1, 1, 1, 1, 1, 1, 1, 1, ...
   1, 2, 1, 0, 1, 0, 1, 2, ...
   1, 2, 0, 1, 2, 0, 1, 2, ...
   1, 3, 1, 1, 1, 1, 1, 3, ...
   1, 2, 0, 0, 2, 1, 2, 0, ...
   1, 4, 0, 0, 2, 0, 1, 4, ...
   1, 2, 2, 0, 2, 0, 0, 1, ...
   1, 4, 1, 0, 1, 0, 1, 4, ...

Seiichi Manyama, Antidiagonals n = 1..100, flattened

Columns k=0..31 give A000012, A000005, A035185, A035186, A001227, A035187, A035188, A035189, A035185, A035191, A035192, A035193, A035194, A035195, A035196, A035197, A001227, A035199, A035200, A035201, A035202, A035203, A035204, A035205, A035188, A035207, A035208, A035186, A035210, A035211, A035212, A035213.

Cf. A215200.

A[n_, k_] := Sum[KroneckerSymbol[k, d], {d, Divisors[n]}];
Table[A[n - k, k], {n, 1, 13}, {k, n - 1, 0, -1}] // Flatten (* Jean-François Alcover, Sep 25 2019 *)

A161528 Expansion of the q-series Sum_{n >= 0} (-1)^nq^(n(n+1)/2)(1-q)(1-q^2)...(1-q^n)/((1-q^(n+1))(1-q^(n+2))...(1-q^(2n+1))).

Original entry on oeis.org

1, 0, 2, 1, 0, 0, 2, 1, 2, 0, 0, 0, 2, 0, 2, 2, 1, 0, 0, 0, 2, 0, 0, 2, 3, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 2, 2, 0, 2, 1, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 2, 1, 2, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2, 2, 0, 0, 2, 0, 4, 0, 0, 2, 3, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 2, 0, 2, 2, 1, 0, 4, 0, 2, 0, 0, 0, 0, 0, 2, 2

Offset: 0

Jeremy Lovejoy, Jun 12 2009

nonn

Jeremy Lovejoy and Olivier Mallet, n-color overpartitions, twisted divisor functions, and Rogers-Ramanujan identities, South East Asian J. Math. Math. Sci., Vol. 6, No. 2 (2008), pp. 23-36.

Cf. A001622, A035187.

f[p_, e_] := If[MemberQ[{2, 3}, Mod[p, 5]], (1 + (-1)^e)/2, e+1]; f[5, e_] := 1; a[0] = 1; a[n_] := Times @@ f @@@ FactorInteger[5*n+1]; Array[a, 100, 0] (* Amiram Eldar, Jan 11 2025 *)

a(n) = {my(f = factor(5*n+1)); prod(i = 1, #f~, if(f[i, 1] == 5, 1, if(f[i, 1] % 5 == 2 || f[i, 1] % 5 == 3, (1 + (-1)^f[i, 2])/2, f[i, 2] + 1)));} \\ Amiram Eldar, Jan 11 2025

a(n) = A035187(5n+1).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4*log(phi)/sqrt(5) = 0.860817..., where phi is the golden ratio (A001622) . - Amiram Eldar, Jan 11 2025

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).

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A370412 Square array T(n, k) = numerator( 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).

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PARI

Sage

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A031366 Number of symmetrically inequivalent coincidence rotations of icosian ring of index n.

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Maple

Formula

Extensions

A300382 Dirichlet series for a cubic module of rank 6.

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Maple

A327785 Square array read by antidiagonals: A(n,k) = Sum_{d|n} (k/d), (n>=1, k>=0), where (m/n) is the Kronecker symbol.

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Mathematica

A161528 Expansion of the q-series Sum_{n >= 0} (-1)^nq^(n(n+1)/2)(1-q)(1-q^2)...(1-q^n)/((1-q^(n+1))(1-q^(n+2))...(1-q^(2n+1))).

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Mathematica

PARI

Formula

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).