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A362728 a(n) = [x^n] E(x)^n where E(x) = exp( Sum_{k >= 1} A108628(k-1)*x^k/k ).

%I A362728 #14 Mar 27 2025 02:24:00
%S A362728 1,1,9,91,985,11101,128475,1515032,18116825,218988046,2669804209,
%T A362728 32776883899,404733925435,5022161428571,62578069656776,
%U A362728 782560813918216,9817011145746649,123492956278927438,1557295053170126994,19681186581532094418
%N A362728 a(n) = [x^n] E(x)^n where E(x) = exp( Sum_{k >= 1} A108628(k-1)*x^k/k ).
%C A362728 A108628(n) = B(n+1,n,n+1) in the notation of Straub, equation 24, where it is shown that the supercongruences A108628(n*p^k) == A108628(n*p^(k-1)) (mod p^(3*k)) hold for all primes p >= 5 and all positive integers n and k.
%C A362728 Inductively define a family of sequences {a(i,n) : n >= 0}, i >= 0, by setting a(0,n) = A108628(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 A362728 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 A362728 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 A362728 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 A362728 A108628 := proc(n) add(binomial(n,k)*binomial(n+1,k)*binomial(n+k+1,k), k = 0..n) end:
%p A362728 E(n,x) := series( exp(n*add(A108628(k-1)*x^k/k, k = 1..20)), x, 21 ):
%p A362728 seq(coeftayl(E(n,x), x = 0, n), n = 0..20);
%Y A362728 Cf. A108628, A362722 - A362733.
%K A362728 nonn,easy
%O A362728 0,3
%A A362728 _Peter Bala_, May 03 2023