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%I #10 May 11 2023 10:25:50
%S 1,2,8,110,960,12502,136952,1746558,20951040,267467294,3347043208,
%T 43051344074,550991269824,7146318966438,92706899799480,
%U 1211369977374310,15857138035286016,208493724775866726,2747100161210031944,36305149229744449050,480750961929272288960
%N a(n) = [x^n] ( E(x)/E(-x) )^n where E(x) = exp( Sum_{k >= 1} A208675(k)*x^k/k ).
%C 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.
%H 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 Conjecture: the supercongruence a(n*p^r) == a(n(p^(r-1)) (mod p^(3*r)) holds for
%F all primes p >= 7 and positive integers n and r.
%p A208675 := proc(n) add( (-1)^k*binomial(n-1,k)*binomial(2*n-k-1,n-k)^2, k = 0..n-1) end:
%p E(n,x) := series(exp(n*add(2*A208675(2*k+1)*x^(2*k+1)/(2*k+1), k = 0..10)), x, 21):
%p seq(coeftayl(E(n,x), x = 0, n), n = 0..20);
%Y Cf. A208675, A362722 - A362733.
%K nonn,easy
%O 0,2
%A _Peter Bala_, May 02 2023
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