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A364298 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.

%I A364298 #8 Jul 20 2023 10:10:40
%S A364298 1,1,1,1,3,19,1,5,73,721,1,7,163,3747,49251,1,9,289,10805,329001,
%T A364298 5370751,1,11,451,23623,1179251,44127003,859748023,1,13,649,43929,
%U A364298 3100001,190464755,8405999785,190320431953,1,15,883,73451,6751251,589050007,42601840975,2160445363107
%N A364298 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.
%C A364298 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.
%C A364298 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
%C A364298 Both types of Apéry numbers satisfy the supercongruences
%C A364298 1) u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r))
%C A364298 and the shifted supercongruences
%C A364298 2) u(n*p^r - 1) == u(n*p^(r-1) - 1) (mod p^(3*r))
%C A364298 for all primes p >= 5 and positive integers n and r.
%C A364298 We conjecture that each row sequence of the present table satisfies the same pair of supercongruences.
%e A364298  Square array begins
%e A364298  n\k|  0    1     2       3          4            5               6
%e A364298   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
%e A364298   1 |  1    1    19     721      49251      5370751       859748023
%e A364298   2 |  1    3    73    3747     329001     44127003      8405999785
%e A364298   3 |  1    5   163   10805    1179251    190464755     42601840975
%e A364298   4 |  1    7   289   23623    3100001    589050007    152184210193
%e A364298   5 |  1    9   451   43929    6751251   1479318759    434790348679
%e A364298   6 |  1   11   649   73451   12953001   3219777011   1062573281785
%p A364298 T(n,k) := coeff(series(1/(1+x)* LegendreP(k,(1-x)/(1+x))^(-n), x, 11), x, k):
%p A364298 # display as a square array
%p A364298 seq(print(seq(T(n, k), k = 0..10)), n = 1..10);
%p A364298 # display as a sequence
%p A364298 seq(seq(T(n-k, k), k = 0..n-1), n = 1..10);
%Y A364298 A364299 (row 1), A364300 (row 2), A364301 (main diagonal), A364302 (first subdiagonal). Cf. A005258, A005259, A143007, A364113.
%K A364298 nonn,tabl,easy
%O A364298 1,5
%A A364298 _Peter Bala_, Jul 18 2023