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A364302
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a(n) = [x^n] 1/(1 + x) * Legendre_P(n, (1 - x)/(1 + x))^(-n-1) for n >= 0.
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4
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1, 3, 163, 23623, 6751251, 3219777011, 2313306332191, 2337707082109071, 3163417897474821763, 5524913023443862515019, 12101947272421487464092429, 32493996621780038121738419591, 104964758754905547830609842389527, 401618040258524641485654323795309235
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OFFSET
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0,2
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COMMENTS
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LINKS
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FORMULA
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Conjectures:
1) the supercongruences a(p) == 2*p + 1 (mod p^3) hold for all primes p >= 5 (checked up to p = 101).
2) the supercongruences a(p - 1) == 1 (mod p^4) hold for all primes p >= 3 (checked up to p = 101).
3) more generally, the supercongruences a(p^k - 1) == 1 (mod p^(3+k)) may hold for all primes p >= 3 and all k >= 1.
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MAPLE
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a(n) := coeff(series( 1/(1 + x) * LegendreP(n, (1 - x)/(1 + x))^(-n-1), x, 21), x, n):
seq(a(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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