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A362731
a(n) = [x^n] E(x)^n where E(x) = exp( Sum_{k >= 1} A000172(k)*x^k/k ).
0
1, 2, 18, 182, 1954, 21702, 246366, 2839846, 33105186, 389264798, 4608481918, 54862022910, 656099844526, 7876525155020, 94867757934870, 1145843922848232, 13873839714404642, 168345900709550388, 2046612356962697502, 24923311881995950740, 303974276349311203854
OFFSET
0,2
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
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.
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);
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Peter Bala, May 05 2023
STATUS
approved