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A364298
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Square array read by ascending antidiagonals: T(n,k) = [x^k] 1/(1 + x) * Legendre_P(k, (1 - x)/(1 + x))^(-n) for n >= 1, k >= 0.
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6
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1, 1, 1, 1, 3, 19, 1, 5, 73, 721, 1, 7, 163, 3747, 49251, 1, 9, 289, 10805, 329001, 5370751, 1, 11, 451, 23623, 1179251, 44127003, 859748023, 1, 13, 649, 43929, 3100001, 190464755, 8405999785, 190320431953, 1, 15, 883, 73451, 6751251, 589050007, 42601840975, 2160445363107
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OFFSET
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1,5
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COMMENTS
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In the square array A364113, the k-th entry in row n is defined as [x^k] 1/(1 - x) * Legendre_P(k, (1 + x)/(1 - x))^n. Here we essentially extend A364113 to negative values of n.
The two types of Apéry numbers A005258 and A005259 are related to the Legendre polynomials by A005258(k) = [x^k] 1/(1 - x) * Legendre_P(k, (1 + x)/(1 - x)) and A005259(k) = [x^k] 1/(1 - x) * Legendre_P(k, (1 + x)/(1 - x))^2 and thus form rows 1 and 2 of the array A364113
Both types of Apéry numbers satisfy the supercongruences
1) u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r))
and the shifted supercongruences
2) u(n*p^r - 1) == u(n*p^(r-1) - 1) (mod p^(3*r))
for all primes p >= 5 and positive integers n and r.
We conjecture that each row sequence of the present table satisfies the same pair of supercongruences.
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LINKS
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EXAMPLE
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Square array begins
n\k| 0 1 2 3 4 5 6
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
1 | 1 1 19 721 49251 5370751 859748023
2 | 1 3 73 3747 329001 44127003 8405999785
3 | 1 5 163 10805 1179251 190464755 42601840975
4 | 1 7 289 23623 3100001 589050007 152184210193
5 | 1 9 451 43929 6751251 1479318759 434790348679
6 | 1 11 649 73451 12953001 3219777011 1062573281785
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MAPLE
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T(n, k) := coeff(series(1/(1+x)* LegendreP(k, (1-x)/(1+x))^(-n), x, 11), x, k):
# display as a square array
seq(print(seq(T(n, k), k = 0..10)), n = 1..10);
# display as a sequence
seq(seq(T(n-k, k), k = 0..n-1), n = 1..10);
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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