OFFSET
0,2
COMMENTS
Symmetric-cube coefficients of the Gauss hypergeometric series at the mixed CM point (1/6,1/3;1).
The sequence has a level-3 modular realization. Let u = eta(3*tau)^12/eta(tau)^12, t = u/(1+27*u)^2, H_mix = q/t, C_0 = 3*E_{5,chi_0,chi_3}, and C_mix = 3*E_{5,chi_0,chi_3} - 27*E_{5,chi_3,chi_0} = (1-27*u)*C_0. Then a(n) = [q^n] C_mix(q)*H_mix(q)^n. Write c_mix(n) = [q^n] C_mix(q).
Theorem 4.33 of Shvets, arXiv:2605.19773, proves that for every split prime p >= 7, p == 1 (mod 3), and every m >= 1, a(m*p) == a(m) (mod p^4). In particular, for split p >= 7, a(p) == 18 (mod p^4).
At the Eisenstein-coefficient level, Theorem 5.1 gives c_mix(m*p^r) == c_mix(m*p^(r-1)) (mod p^(4*r)) for p >= 5, chi_3(p) = 1, and m,r >= 1.
The same paper proves an inert-prime obstruction. Theorem 5.2 states that for p >= 5 with chi_3(p) = -1, beta(p)-beta(1) = p^4-2 is not 0 (mod p). More generally, if m = p^a*m_0, p does not divide m_0, and beta(m_0) is not 0 (mod p), then v_p(c_mix(m*p)-c_mix(m)) = 0. Here beta(n) = Sum_{d|n} chi_3(n/d)*d^4.
Proposition 5.3 says that for inert p >= 5 the split-style formal q-congruence Lambda_p(C_mix*H_mix^(p*X)) == C_mix*H_mix^X (mod (p^4, X^4)) cannot hold.
For eps = +-1 define A_eps(m) = [q^m] (C_0(q)-27*eps*u(q)*C_0(q))*H_mix(q)^m. Then A_+1(m) = a(m). Theorem 5.4 gives the first-digit Cartier parity law a(m*p^r) == A_eps(m) (mod p), where eps = chi_3(p)^r, for every prime p >= 5, every r >= 0, and every m >= 0.
Consequently, for inert primes p >= 5, p == 2 (mod 3), a(m*p^(2*r)) == a(m) (mod p), and a(m*p^(2*r+1)) == A_-1(m) (mod p). Corollary 5.5 gives a(p) == 72 (mod p) for every inert prime p >= 5. Since a(1) = 18, the split-style congruence fails already modulo p at inert primes.
LINKS
Frits Beukers, Supercongruences using modular forms, arXiv:2403.03301 [math.NT], 2024-2025.
Vaclav Kotesovec, Graph - the asymptotic ratio (1000 terms).
L. Long, F.-T. Tu, N. Yui, and W. Zudilin, Supercongruences for rigid hypergeometric Calabi-Yau threefolds, Adv. Math. 393 (2021), Paper No. 108058.
Alex Shvets, Split-prime supercongruence at the mixed CM point (1/6, 1/3; 1), arXiv:2605.19773 [math.NT], 2026.
FORMULA
G.f.: hypergeom([1/6, 1/3], [1], 108*z)^3.
a(n) satisfies the order-2 recurrence (n+2)^4*a(n+2) - 6*(36*n^4 + 198*n^3 + 424*n^2 + 417*n + 158)*a(n+1) + 324*(n+1)*(2*n+1)*(3*n+2)*(6*n+5)*a(n) = 0, with a(0)=1, a(1)=18.
With C_mix and H_mix as in Comments, a(n) = [q^n] C_mix(q)*H_mix(q)^n.
a(n) ~ Gamma(1/3)^3 * 2^(2*n - 13/3) * 3^(3*n + 5/2) / (Pi^(7/2) * n^(3/2)). - Vaclav Kotesovec, May 08 2026
EXAMPLE
a(0) = 108^0 * [z^0] (1 + z/18 + 7*z^2/324 + ...)^3 = 1.
a(1) = 108 * [z^1] (1 + z/18 + 7*z^2/324 + ...)^3 = 108 * 1/6 = 18.
a(2) = 108^2 * [z^2] (1 + z/18 + 7*z^2/324 + ...)^3 = 11664 * 2/27 = 864.
MATHEMATICA
CoefficientList[Series[Hypergeometric2F1[1/6, 1/3, 1, 108 z]^3, {z, 0, 15}], z]
PROG
(PARI) a(n) = {my(A=vector(max(n+1, 2))); A[1]=1; A[2]=18; for(k=0, n-2, A[k+3] = (6*(36*k^4+198*k^3+424*k^2+417*k+158)*A[k+2] - 324*(k+1)*(2*k+1)*(3*k+2)*(6*k+5)*A[k+1])/(k+2)^4); A[n+1]}
(PARI) Vec(hypergeom([1/6, 1/3], [1], 108*z)^3) \\ Michel Marcus, May 13 2026
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Alex Shvets, May 07 2026
STATUS
approved