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A362726 a(n) = [x^n] E(x)^n where E(x) = exp( Sum_{k >= 1} A208675(k)*x^k/k ).

%I A362726 #16 Mar 27 2025 02:23:51
%S A362726 1,1,7,64,647,6901,76120,859216,9863303,114689746,1347186307,
%T A362726 15954752903,190235245976,2281177393704,27487043703672,
%U A362726 332588768198389,4038905184944263,49204502405466061,601135759955624038,7362647062772162397,90380912127647103747
%N A362726 a(n) = [x^n] E(x)^n where E(x) = exp( Sum_{k >= 1} A208675(k)*x^k/k ).
%C A362726 A208675(n) = B(n,n-1,n-1) in the notation of Straub, equation 24, where it is shown that the supercongruences A208675(n*p^k) == A208675(n*p^(k-1)) (mod p^(3*k)) hold for all primes p >= 5 and all positive integers n and k.
%C A362726 Inductively define a family of sequences {a(i,n) : n >= 0}, i >= 0, by setting a(0,n) = A208675(n) and, for i >= 1, a(i,n) = [x^n] ( exp(Sum_{k >= 1} a(i-1,k)*x^k/k) )^n. In this notation the present sequence is {a(1,n)}.
%C A362726 We conjecture that the sequences {a(i,n) : n >= 0}, i >= 1, satisfy the supercongruences u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r)) for all primes p >= 7, and positive integers n and r.
%H A362726 Armin Straub, <a href="http://dx.doi.org/10.2140/ant.2014.8.1985">Multivariate Apéry numbers and supercongruences of rational functions</a>, Algebra & Number Theory, Vol. 8, No. 8 (2014), pp. 1985-2008; <a href="https://arxiv.org/abs/1401.0854">arXiv preprint</a>, arXiv:1401.0854 [math.NT], 2014.
%F A362726 Conjecture: the supercongruence a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) holds for all primes p >= 7 and positive integers n and r.
%p A362726 A208675 := proc(n) add( (-1)^k*binomial(n-1,k)*binomial(2*n-k-1,n-k)^2, k = 0..n-1) end:
%p A362726 E(n,x) := series(exp(n*add(A208675(k)*x^k/k, k = 1..20)), x, 21):
%p A362726 seq(coeftayl(E(n,x), x = 0, n), n = 0..20);
%Y A362726 Cf. A208675, A362722 - A362733.
%K A362726 nonn,easy
%O A362726 0,3
%A A362726 _Peter Bala_, May 02 2023