OFFSET
0,2
COMMENTS
The sequence appears in several series converging to 1/Pi, called Ramanujan(-Sato) type series and constructed from modular forms - in this case, modular forms of level 23.
Analogous theory for more levels is covered in the book "Ramanujan's Theta Functions" by S. Cooper.
This is sequence c_{23} in the 2015 paper of Cooper et al.
Apéry-like sequence (The term "Apery-like" is not well-defined).
An explicit binomial formula is not known.
LINKS
X. Caruso, F. Fürnsinn, D. Vargas-Montoya, and W. Zudilin, Galois Groups of Apéry-like Series Modulo Primes, arXiv:2510.23298 [math.NT], 2025.
S. Cooper, Ramanujan’s Theta Functions, Springer International Publishing, (2017).
S. Cooper, J. Ge, and D. Ye. Hypergeometric Transformation Formulas of Degrees 3, 7, 11 and 23, Journal of Mathematical Analysis and Applications 421 (2) (2015), pp. 1358-76.
FORMULA
(n + 1)^3*a(n + 1) = (2*n + 1)*(7*n^2 + 7*n + 4)*a(n) - n*(57*n^2 + 31)*a(n - 1) + (2*n - 1)*(53*n^2 - 53*n + 42)*a(n - 2) - 6*(n - 1)*(15*n^2 - 30*n + 19)*a(n - 3) + 8*(n - 1)*(n - 2)*(2*n - 3)*a(n - 4) + 19*(n - 1)*(n - 2)*(n - 3)*a(n - 5) with a(n)=0 for n<0 and a(0)=1.
G.f.: f(x) satisfies x^2*(1 - 11*x + 22*x^2 - 19*x^3) * (1 - 3*x + 2*x^2 + x^3)*f'''(x)+ 3*x*(1 - 21*x + 114*x^2 - 265*x^3 + 270*x^4 - 56*x^5 - 76*x^6)*f''(x) + (1 - 50*x + 430*x^2 - 1356*x^3 + 1734*x^4 - 432*x^5 - 684*x^6)*f'(x) - 4*(1 - 22*x + 111*x^2 - 192*x^3 + 60*x^4 + 114*x^5)*f(x) = 0.
PROG
(SageMath)
@cached_function
def T23(n):
if n < 0:
return QQ(0)
if n== 0:
return QQ(1)
Tn = (14*n^3 - 21*n^2 + 15*n - 4) * T23(n-1)
Tn -= (57*n^3 - 171*n^2 + 202*n - 88) * T23(n-2)
Tn += (106*n^3 - 477*n^2 + 773*n - 444) * T23(n-3)
Tn -= (90*n^3 - 540*n^2 + 1104*n - 768) * T23(n-4)
Tn += (16*n^3 - 120*n^2 + 296*n - 240) * T23(n-5)
Tn += (19*n^3 - 171*n^2 + 494*n - 456) * T23(n-6)
Tn /= n^3
return Tn
CROSSREFS
KEYWORD
nonn
AUTHOR
Florian Fürnsinn, Oct 30 2025
STATUS
approved