A362731 a(n) = [x^n] E(x)^n where E(x) = exp( Sum_{k >= 1} A000172(k)*x^k/k ).

1, 2, 18, 182, 1954, 21702, 246366, 2839846, 33105186, 389264798, 4608481918, 54862022910, 656099844526, 7876525155020, 94867757934870, 1145843922848232, 13873839714404642, 168345900709550388, 2046612356962697502, 24923311881995950740, 303974276349311203854

Offset: 0

Author

Peter Bala, May 05 2023

Keywords

• nonn
• easy

Comments

It is known that the sequence of Franel numbers A000172 satisfies the Gauss congruences A000172(n*p^r) == A000172(n*p^(r-1)) (mod p^r) for all primes p and positive integers n and r.

One consequence is that the power series expansion of E(x) = exp( Sum_{k >= 1} A000172(k)*x^k/k ) = 1 + 2*x + 7*x^2 + 30*x^3 + 147*x^4 + ... (the g.f. of A166990) has integer coefficients (see, for example, Beukers, Proposition, p. 143). Therefore a(n) = [x^n] E(x)^n is an integer.

In fact, the Franel numbers satisfy stronger congruences than the Gauss congruences known as supercongruences: A000172(n*p^r) == A000172(n*p^(r-1)) (mod p^(3*r)) for all primes p >= 5 and positive integers n and r.

Links

• F. Beukers, Some congruences for the Apery numbers, Journal of Number Theory, Vol. 21, Issue 2, Oct. 1985, pp. 141-155. local copy

Crossrefs

Cf. A000172, A166990, A362722 - A362733.

Programs

Maple

A000172 := proc(n) add(binomial(n,k)^3, k = 0..n); end:
E(n,x) := series( exp(n*add(A000172(k)*x^k/k, k = 1..20)), x, 21 ):
seq(coeftayl(E(n,x), x = 0, n), n = 0..20);

Formula

The Gauss congruence a(n*p^r) == a(n*p^(r-1)) (mod p^r) holds for all primes p and positive integers n and r.

Conjecture: the supercongruence a(n*p^r) == a(n*p^(r-1)) (mod p^(2*r)) holds for

all primes p and positive integers n and r.