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

1, 0, 0, 9, 16, 0, 171, 539, 528, 3654, 16500, 29282, 101851, 483340, 1215445, 3416634, 14564880, 44585475, 124007202, 462804166, 1555048516, 4547401595, 15500748802, 53459717443, 164998563675, 538593687500, 1845162146828, 5920282930815, 19091999953749, 64389113743812, 211137579083046

Offset: 0

Author

Peter Bala, Apr 22 2024

Keywords

• nonn
• easy

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.

Examples

			Supercongruences:
a(11) = 29282 = 2*(11^4) == 0 (mod 11^4).
a(13) = 483340 = (2^2)*5*11*(13^3) == 0 (mod 13^3).
a(2*11) = 15500748802 = 2*7*(11^4)*47*1609 == 0 (mod 11^4).
		

References

• R. P. Stanley, Enumerative Combinatorics Volume 2, Cambridge Univ. Press, 1999, Theorem 6.33, p. 197.

Crossrefs

Cf. A351858, A372211, A372212.

Programs

Maple

f(x) := (1 - x^7)^7/((1 - x^3)^3*(1 - x^4)^4):
seq(coeftayl(f(x)^n, x = 0, n), n = 0..30);

Formula

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.

Extensions

a(28) corrected by and more terms from Georg Fischer, Jul 28 2025