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A362727
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a(n) = [x^n] ( E(x)/E(-x) )^n where E(x) = exp( Sum_{k >= 1} A208675(k)*x^k/k ).
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0
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1, 2, 8, 110, 960, 12502, 136952, 1746558, 20951040, 267467294, 3347043208, 43051344074, 550991269824, 7146318966438, 92706899799480, 1211369977374310, 15857138035286016, 208493724775866726, 2747100161210031944, 36305149229744449050, 480750961929272288960
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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FORMULA
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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.
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MAPLE
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A208675 := proc(n) add( (-1)^k*binomial(n-1, k)*binomial(2*n-k-1, n-k)^2, k = 0..n-1) end:
E(n, x) := series(exp(n*add(2*A208675(2*k+1)*x^(2*k+1)/(2*k+1), k = 0..10)), x, 21):
seq(coeftayl(E(n, x), x = 0, n), n = 0..20);
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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