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

0,3

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

Table of n, a(n) for n=0..25.

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

Prog

(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

Crossrefs

Cf. A224529 (sequence f_n).

Cf. A028930, A030199.

Sequence in context: A073969 A379724 A295863 * A120950 A055695 A113593

Adjacent sequences: A224527 A224528 A224529 * A224531 A224532 A224533

Keyword

nonn

Author

Joerg Arndt, Apr 09 2013

Extensions

More terms from Robin Visser, Aug 03 2023

Status

approved