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A362729 a(n) = [x^n] ( E(x)/E(-x) )^n where E(x) = exp( Sum_{k >= 1} A108628(k-1)*x^k/k ).

%I A362729 #11 Mar 27 2025 02:24:06
%S A362729 1,2,8,146,1344,18502,214136,2820834,35377152,465110894,6038588808,
%T A362729 79936149174,1056557893440,14094461001558,188319357861944,
%U A362729 2529143690991946,34042038343081984,459723572413090934,6221522287903354568,84397945280561045302,1147007337762078241344
%N A362729 a(n) = [x^n] ( E(x)/E(-x) )^n where E(x) = exp( Sum_{k >= 1} A108628(k-1)*x^k/k ).
%C A362729 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.
%H A362729 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 A362729 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 A362729 A108628 := proc(n) add(binomial(n,k)*binomial(n+1,k)*binomial(n+k+1,k) , k = 0..n) end:
%p A362729 E(n,x) := series(exp(n*add(2*(A108628(2*k)*x^(2*k+1))/(2*k+1), k = 0..10)), x, 21):
%p A362729 seq(coeftayl(E(n,x), x = 0, n), n = 0..20);
%Y A362729 Cf. A108628, A362722 - A362733.
%K A362729 nonn,easy
%O A362729 0,2
%A A362729 _Peter Bala_, May 03 2023