A224530 Sequence F_n from a paper by Robert Osburn and Brundaban Sahu.

1, 0, 2, 6, 30, 144, 758, 4080, 22702, 128832, 744300, 4359972, 25842414, 154689912, 933828324, 5678696556, 34754244174, 213901762464, 1323104558204, 8220846355956, 51284447272084, 321095305733280, 2017050339848388, 12708912192988128, 80296949632284814, 508618518515268720

Offset: 0

Author

Joerg Arndt, Apr 09 2013

Keywords

• nonn

Comments

These are the coefficients of the power series expansion of F with respect to powers of t_2, where F(z) = Sum_{k,l in Z} q^(2*k^2 + k*l + 3*l^2) and t_2(z) = eta(z)*eta(23*z)/F(z), where eta(z) is the Dedekind eta-function and q = exp(2*Pi*i*z). - Robin Visser, Aug 03 2023

Osburn and Sahu prove that if p is a prime which is a quadratic residue mod 23 and n, r are positive integers, then a(n*p^r) == a(n*p^(r-1)) (mod p). - Robin Visser, Aug 03 2023

Links

• Robert Osburn and Brundaban Sahu, Congruences via modular forms, arXiv:0912.0173 [math.NT], 2009-2010.

Crossrefs

Cf. A224529 (sequence f_n).

Cf. A028930, A030199.

Programs

Sage

def a(n):
    if n==0: return 1
    F=sum([sum([x^(2*a^2+a*b+3*b^2) for a in range(-n,n)]) for b in range(-n,n)])
    eta = x^(1/24)*product([(1 - x^k) for k in range(1, n)])
    t2 = eta*eta(x=x^23)/F
    for k in range(1, n):
        c = F.taylor(x, 0, k).coefficient(x^k)
        F -= c*(t2^k)
    return F.taylor(x, 0, n).coefficient(x^n)  # Robin Visser, Aug 03 2023

Extensions

More terms from Robin Visser, Aug 03 2023