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

%I A362731 #11 Mar 27 2025 05:43:39
%S A362731 1,2,18,182,1954,21702,246366,2839846,33105186,389264798,4608481918,
%T A362731 54862022910,656099844526,7876525155020,94867757934870,
%U A362731 1145843922848232,13873839714404642,168345900709550388,2046612356962697502,24923311881995950740,303974276349311203854
%N A362731 a(n) = [x^n] E(x)^n where E(x) = exp( Sum_{k >= 1} A000172(k)*x^k/k ).
%C A362731 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.
%C A362731 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.
%C A362731 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.
%H A362731 F. Beukers, <a href="https://doi.org/10.1016/0022-314X(85)90047-2">Some congruences for the Apery numbers</a>, Journal of Number Theory, Vol. 21, Issue 2, Oct. 1985, pp. 141-155. <a href="/A339710/a339710.pdf">local copy</a>
%F A362731 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.
%F A362731 Conjecture: the supercongruence a(n*p^r) == a(n*p^(r-1)) (mod p^(2*r)) holds for
%F A362731 all primes p and positive integers n and r.
%p A362731 A000172 := proc(n) add(binomial(n,k)^3, k = 0..n); end:
%p A362731 E(n,x) := series( exp(n*add(A000172(k)*x^k/k, k = 1..20)), x, 21 ):
%p A362731 seq(coeftayl(E(n,x), x = 0, n), n = 0..20);
%Y A362731 Cf. A000172, A166990, A362722 - A362733.
%K A362731 nonn,easy
%O A362731 0,2
%A A362731 _Peter Bala_, May 05 2023