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A362726
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a(n) = [x^n] 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, 1, 7, 64, 647, 6901, 76120, 859216, 9863303, 114689746, 1347186307, 15954752903, 190235245976, 2281177393704, 27487043703672, 332588768198389, 4038905184944263, 49204502405466061, 601135759955624038, 7362647062772162397, 90380912127647103747
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OFFSET
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0,3
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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.
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)}.
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.
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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(A208675(k)*x^k/k, k = 1..20)), 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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