OFFSET
0,3
COMMENTS
Let G(x) be a formal power series with integer coefficients. The sequence defined by g(n) = [x^n] G(x)^n satisfies the Gauss congruences: g(n*p^r) == g(n*p^(r-1)) (mod p^r) for all primes p and positive integers n and r.
We conjecture that in this case the stronger supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) hold for primes p >= 11 and positive integers n and r. Some examples are given below. Cf. A351858.
More generally, if r is a positive integer and s an integer then the sequence defined by u(r,s; n) = [x^(r*n)] f(x)^(s*n) may satisfy the same supercongruences.
REFERENCES
R. P. Stanley, Enumerative Combinatorics Volume 2, Cambridge Univ. Press, 1999, Theorem 6.33, p. 197.
FORMULA
The o.g.f. A(x) = 1 + 4*x^2 + 36*x^4 + 25*x^5 + ... is the diagonal of the bivariate rational function 1/(1 - t*f(x)) and hence is an algebraic function over the field of rational functions Q(x) by Stanley, Theorem 6.33, p. 197.
EXAMPLE
Supercongruences:
a(11) = 91839 = 3*(11^3)*23 == 0 (mod 11^3).
a(2*11) - a(2) = 154437142545 - 4 = (11^3)*2671*43441 == 0 (mod 11^3).
MAPLE
f(x) := (1 - x^7)^7/((1 - x^2)^2*(1 - x^5)^5):
seq(coeftayl(f(x)^n, x = 0, n), n = 0..27);
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Peter Bala, Apr 22 2024
STATUS
approved