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%I #12 Nov 04 2025 15:34:12
%S 1,4,16,76,424,2632,17464,121096,866440,6347656,47373448,358877296,
%T 2752367704,21328244080,166734121384,1313368255504,10413961927432,
%U 83054919474448,665804730685672,5361910101292144
%N Coefficients in level 23 Ramanujan-Sato series for 1/Pi.
%C 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.
%C Analogous theory for more levels is covered in the book "Ramanujan's Theta Functions" by S. Cooper.
%C This is sequence c_{23} in the 2015 paper of Cooper et al.
%C Apéry-like sequence (The term "Apery-like" is not well-defined).
%C An explicit binomial formula is not known.
%H X. Caruso, F. Fürnsinn, D. Vargas-Montoya, and W. Zudilin, <a href="https://arxiv.org/abs/2510.23298">Galois Groups of Apéry-like Series Modulo Primes</a>, arXiv:2510.23298 [math.NT], 2025.
%H S. Cooper, <a href="https://doi.org/10.1007/978-3-319-56172-1">Ramanujan’s Theta Functions</a>, Springer International Publishing, (2017).
%H S. Cooper, J. Ge, and D. Ye. <a href="https://doi.org/10.1016/J.JMAA.2014.07.061">Hypergeometric Transformation Formulas of Degrees 3, 7, 11 and 23</a>, Journal of Mathematical Analysis and Applications 421 (2) (2015), pp. 1358-76.
%F (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.
%F 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.
%o (SageMath)
%o @cached_function
%o def T23(n):
%o if n < 0:
%o return QQ(0)
%o if n== 0:
%o return QQ(1)
%o Tn = (14*n^3 - 21*n^2 + 15*n - 4) * T23(n-1)
%o Tn -= (57*n^3 - 171*n^2 + 202*n - 88) * T23(n-2)
%o Tn += (106*n^3 - 477*n^2 + 773*n - 444) * T23(n-3)
%o Tn -= (90*n^3 - 540*n^2 + 1104*n - 768) * T23(n-4)
%o Tn += (16*n^3 - 120*n^2 + 296*n - 240) * T23(n-5)
%o Tn += (19*n^3 - 171*n^2 + 494*n - 456) * T23(n-6)
%o Tn /= n^3
%o return Tn
%Y Cf. A005259, A002895, A125143, A290575, A290576, A274786, A181418, A183204, A005260, A284756, A219692, A386283, A386708.
%K nonn
%O 0,2
%A _Florian Fürnsinn_, Oct 30 2025