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A224530 Sequence F_n from a paper by Robert Osburn and Brundaban Sahu. 2
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 (list; graph; refs; listen; history; text; internal format)
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
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).
Sequence in context: A035105 A073969 A295863 * A120950 A055695 A113593
KEYWORD
nonn
AUTHOR
Joerg Arndt, Apr 09 2013
EXTENSIONS
More terms from Robin Visser, Aug 03 2023
STATUS
approved

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Last modified April 22 12:06 EDT 2024. Contains 371900 sequences. (Running on oeis4.)