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A372213 a(n) = [x^n] f(x)^n, where f(x) = (1 - x^7)^7/((1 - x^3)^3 * (1 - x^4)^4).

%I A372213 #12 Jul 28 2025 13:50:56
%S A372213 1,0,0,9,16,0,171,539,528,3654,16500,29282,101851,483340,1215445,
%T A372213 3416634,14564880,44585475,124007202,462804166,1555048516,4547401595,
%U A372213 15500748802,53459717443,164998563675,538593687500,1845162146828,5920282930815,19091999953749,64389113743812,211137579083046
%N A372213 a(n) = [x^n] f(x)^n, where f(x) = (1 - x^7)^7/((1 - x^3)^3 * (1 - x^4)^4).
%C A372213 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.
%C A372213 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.
%C A372213 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.
%D A372213 R. P. Stanley, Enumerative Combinatorics Volume 2, Cambridge Univ. Press, 1999, Theorem 6.33, p. 197.
%F A372213 The o.g.f. A(x) = 1 + 9*x^3 + 16*x^4 + 171*x^6 + ... 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.
%e A372213 Supercongruences:
%e A372213 a(11) = 29282 = 2*(11^4) == 0 (mod 11^4).
%e A372213 a(13) = 483340 = (2^2)*5*11*(13^3) == 0 (mod 13^3).
%e A372213 a(2*11) = 15500748802 = 2*7*(11^4)*47*1609 == 0 (mod 11^4).
%p A372213 f(x) := (1 - x^7)^7/((1 - x^3)^3*(1 - x^4)^4):
%p A372213 seq(coeftayl(f(x)^n, x = 0, n), n = 0..30);
%Y A372213 Cf. A351858, A372211, A372212.
%K A372213 nonn,easy
%O A372213 0,4
%A A372213 _Peter Bala_, Apr 22 2024
%E A372213 a(28) corrected by and more terms from _Georg Fischer_, Jul 28 2025