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A364117
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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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3
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1, 5, 163, 14409, 2511251, 730485013, 320259339415, 197591579213969, 163325387776051459, 174310058440646865021, 233402385203650889753429, 383208210107883180333696265, 757120215942256247847040802463, 1772210276849283299764079883683173
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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 + 3 (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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