Supercongruences modulo p^(3k) for the coefficients of powers of (1+x)^A/(1-x)^B: a proof of P. Bala's conjectures on the Schröder-type sequences A333715, A103885 and A333481
Year: 2026
Type: preprint
Abstract: For integers r, A, B ≥ 1 let u(N) = [x^(rN)] ((1+x)^A (1−x)^(−B))^N = Sum_i C(AN, i) C((B+r)N − i − 1, BN − 1). We prove that for every prime p ≥ 5 and all n, k ≥ 1, u(n p^k) ≡ u(n p^(k−1)) (mod p^(3k)). This settles the supercongruence conjectures of P. Bala in the OEIS for A333715 ([x^(3n)]((1+x)/(1−x))^n, April 2020), A103885 ([x^n] S(x)^n, S the large Schröder numbers) and A333481 ([x^n] S(x)^(2n), March 2020), and the same congruence for A002003, with the exponent 3k attained. The proof splits the sum defining u(n p^k) according to the residue of i modulo p: the terms with p | i are handled by the Jacobsthal–Kazandzidis congruence; the remaining terms are written exactly as T(pa) p^2 (−1)^(b−1) c e /((pa+b)(pd−b)) times a p-adic unit, the sum over b is expanded about the centre b = p/2, where Sum_(b=1..p−1) (−1)^b (b − p/2)^(−2j) = 0, and the surviving terms are controlled by the congruence Sum_a w(a) (2a+1)^(2j+1) ≡ 0 (mod N) for the odd moments of the weights w(a) = C(AN−1, a) C((B+r)N − a − 1, BN), which we prove for every odd integer N by a derivative identity for (1+z)^α (1−z)^(−β) modulo N. The Catalan-type conjectures of Bala (A333093–A333097, A333564) are not of this form and are not covered. The conjectures were selected by a scan of a local copy of the OEIS for open "tower" congruences with the author's SyntheticMind engine, which had already proved the towers of Euler, Genocchi and tribonacci type; the method (recognition of the form λ·[x^(rn)]((1+x)^A/(1−x)^B)^n, exact check and proof) has been implemented in the engine. The proofs and the text were prepared with the assistance of an AI system under the author's direction; the author checked every step and takes full responsibility for the content. Files: Blanco_Gomez_2026_supercongruences_binomial_powers_EN.pdf is the article (6 pages); Blanco_Gomez_2026_supercongruencias_potencias_binomiales_ES.pdf is the Spanish version (same content); verify_super_towers.py is an independent verification script (Python 3, exact integers and fractions, standard library only) that checks the theorem with the exact valuation for eight triples (r, A, B), primes 5 ≤ p ≤ 19 and k ≤ 4, the closed form as an exact identity, the Jacobsthal–Kazandzidis congruence, the odd-moment theorem and the key congruence of its proof (4256 cases), and the centred alternating sums; verify_super_towers.log is its output (ALL CHECKS PASSED).
Author: Blanco Gómez, Roberto
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7222754653
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