Roberto Blanco Gómez

Supercongruences modulo p^(3k) and integrality for factorial ratios with fractional arguments, with proofs of conjectures of P. Bala

Let a(N) = c ∏ Γ(α_i N + 1)^(e_i) with rational α_i > 0 of common denominator d, integers e_i and a rational constant c. Assume that Σ e_i α_i = 0 and that, for every N, the factors whose arguments α_i N have the same non-zero fractional part have total exponent zero. We prove th...

Kummer congruences for the values b^r E_r(a/b) of the Euler polynomials at rational points, and

Let E_r(x) be the Euler polynomials and let p be an odd prime. We prove that for every rational a/b with p not dividing b the numbers f_r = b^r E_r(a/b) are p-integral and satisfy f_r == f_r' (mod p^k) whenever r == r' (mod (p-1) p^(k-1)) and r, r' >= k; this is the only hypothes...

Supercongruences modulo p^(3k) for [x^(rn)]((1+x)^α(1−x)^(−β))^n with arbitrary integer exponents, and proofs of conjectures of P. Bala on twenty sequences and four infinite families

For integers r ≥ 1 and α, β let u(N) = [x^(rN)] ((1+x)^α (1−x)^(−β))^N. We prove that for every prime p ≥ 5, all integers α, β (of either sign) and all n, k ≥ 1, u(n p^k) ≡ u(n p^(k−1)) (mod p^(3k+v)), where v = min(v_p(α), v_p(β)). For α, β ≥ 1 and without the factor p^v this wa...

A proof of two congruences of P. Bala for the Motzkin numbers modulo p and p^2, with a generalization to coefficients of powers of a polynomial

We prove two congruences conjectured by P. Bala in the OEIS entry A001006 of the Motzkin numbers M(n) (February 10, 2022): M(n+p^k) == M(n) (mod p) for every prime p >= 3, k >= 1 and 0 <= n <= p^k - 3, and M(n+p^k) == M(n) (mod p^2) for every prime p >= 5, k >= 2 and 0 <= n <= p^...

A proof of a conjecture of H. Gruber: finite unary languages by the alphabetic width of their regular expressions

We prove a statement recorded by H. Gruber in 2012 in the OEIS entry A000079 (the powers of 2), verified by him up to n = 17: for n >= 1, the number of distinct finite languages over a one-letter alphabet whose minimum regular expression has alphabetic width n is 2^n. The alphabe...

The fixed space of the ribbon-to-homogeneous transition matrix of noncommutative symmetric functions: a proof of a conjecture recorded by J. M. Campbell

We prove a conjecture recorded by J. M. Campbell in 2018 in the OEIS entry A001405 (the central binomial coefficients binomial(n, floor(n/2))). Let NSym be the algebra of noncommutative symmetric functions of Gelfand, Krob, Lascoux, Leclerc, Retakh and Thibon, and let C_n(R, H) b...

A proof of a conjecture of H. Gruber: finite unary languages by the alphabetic width of their regular expressions

We prove a statement recorded by H. Gruber in 2012 in the OEIS entry A000079 (the powers of 2), verified by him up to n = 17: for n >= 1, the number of distinct finite languages over a one-letter alphabet whose minimum regular expression has alphabetic width n is 2^n. The alphabe...

The fixed space of the ribbon-to-homogeneous transition matrix of noncommutative symmetric functions: a proof of a conjecture recorded by J. M. Campbell

We prove a conjecture recorded by J. M. Campbell in 2018 in the OEIS entry A001405 (the central binomial coefficients binomial(n, floor(n/2))). Let NSym be the algebra of noncommutative symmetric functions of Gelfand, Krob, Lascoux, Leclerc, Retakh and Thibon, and let C_n(R, H) b...

A proof of a conjecture of T. Piesk: nim-products of powers of 2 and Stern's diatomic sequence

We prove a conjecture of T. Piesk stated in 2013 in the OEIS entries A002487 (Stern's diatomic sequence) and A223541 (the array of nim-products 2^m ⊗ 2^n): for every n >= 0, the number of distinct values of 2^x ⊗ 2^y with x + y = n equals A002487(n+1), the number of hyperbinary r...

A proof of a conjecture of T. Piesk: nim-products of powers of 2 and Stern's diatomic sequence

We prove a conjecture of T. Piesk stated in 2013 in the OEIS entries A002487 (Stern's diatomic sequence) and A223541 (the array of nim-products 2^m ⊗ 2^n): for every n >= 0, the number of distinct values of 2^x ⊗ 2^y with x + y = n equals A002487(n+1), the number of hyperbinary r...

A proof of a conjecture of P. Bala on the squares of the Apéry numbers for ζ(2)

We prove a conjecture of P. Bala (OEIS A005258, July 8, 2024): for every n >= 0, a(n)^2 = Sum_{k=0..n} (-1)^(n+k) C(n,k) C(n+k,k) T(n,k), where a(n) = Sum_k C(n,k)^2 C(n+k,k) are the Apéry numbers associated with zeta(2) (A005258) and T(n,k) = Sum_j C(n,j) C(n+j,j) C(k,j) C(k+j,j...