A386708 - OEIS

A386708

Coefficients in level 20 Ramanujan-Sato series for 1/Pi.

1, 4, 20, 120, 820, 6128, 48712, 404480, 3467380, 30449680, 272490096, 2475669984, 22773747400, 211687194400, 1985153038400, 18758489742400, 78433735606900, 1707187621512400, 16418042200744400, 158619019424898400, 1538786416105372400, 14983597493316174400

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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 20.

Analogous theory for levels different from 20 is covered in the book "Ramanujan's Theta Functions" by S. Cooper.

The generating function is Z(X) in the 2018 paper of Huber et al.

LINKS

Table of n, a(n) for n=0..21.

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).

T. Huber, D. Schultz, and D. Ye, Series for 1/Pi of level 20, arXiv:1711.00456 [math.NT], 2017; Journal of Number Theory, 188 (2018), pp. 121-136. DOI: 10.1016/j.jnt.2017.12.010

FORMULA

(n + 1)^3*a(n + 1) = 4*(2*n + 1)*(2*n^2 + 2*n + 1)*a(n) - 16*n*(4*n^2 + 1)*a(n - 1) + 8*(2*n - 1)^3*a(n - 2) with a(n)=0 for n<0 and a(0)=1.

G.f.: f(x) satisfies x^2*(1 - 4*x)*(1 - 12*x + 16*x^2)*f'''(x) + 3*x*(1 - 24*x + 128*x^2 - 160*x^3) * f''(x) + (1 - 56*x + 464*x^2 - 784*x^3)*f'(x) - 4*(1 - 20*x + 54*x^2)*f(x)=0.

PROG

(SageMath)

@cached_function

def T20(n):

if n < 0:

return 0

if n == 0:

return 1

m = n - 1

Tn = 4 * (2*m + 1) * (2*m^2 + 2*m + 1) * T20(n-1)

Tn -= 16 * m * (4*m^2 + 1) * T20(n-2)

Tn += 8 * (2*m - 1)^3 * T20(n-3)