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%I #17 Jul 22 2023 21:19:56
%S 1,1,1,1,3,1,1,5,19,1,1,7,73,147,1,1,9,163,1445,1251,1,1,11,289,5623,
%T 33001,11253,1,1,13,451,14409,235251,819005,104959,1,1,15,649,29531,
%U 908001,11009257,21460825,1004307,1,1,17,883,52717,2511251,65898009,554159719,584307365,9793891,1
%N Square array read by ascending antidiagonals: T(n,k) = [x^k] 1/(1 - x) * Legendre_P(k, (1 + x)/(1 - x))^n for n, k >= 0.
%C 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 present array.
%C Both types of Apéry numbers satisfy the supercongruences
%C 1) u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r))
%C and the shifted supercongruences
%C 2) u(n*p^r - 1) == u(n*p^(r-1) - 1) (mod p^(3*r))
%C for all primes p >= 5 and positive integers n and r.
%C We conjecture that each row sequence of the present table satisfies the same pair of supercongruences.
%e Square array begins
%e n\k| 0 1 2 3 4 5 6 7
%e - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
%e 0 | 1 1 1 1 1 1 1 1
%e 1 | 1 3 19 147 1251 11253 104959 1004307
%e 2 | 1 5 73 1445 33001 819005 21460825 584307365
%e 3 | 1 7 163 5623 235251 11009257 554159719 29359663991
%e 4 | 1 9 289 14409 908001 65898009 5246665201 445752724041
%e 5 | 1 11 451 29531 2511251 251831261 28224521263 3423024241627
%e 6 | 1 13 649 52717 5665001 730485013 106898093065 17144295476461
%p T(n,k) := coeff(series(1/(1-x)* LegendreP(k,(1+x)/(1-x))^n, x, 11), x, k):
%p # display as a square array
%p seq(print(seq(T(n, k), k = 0..10)), n = 0..10);
%p # display as a sequence
%p seq(seq(T(n-k, k), k = 0..n), n = 0..10);
%Y Cf. A005258 (row 1), A005259 (row 2), A364114 (row 3), A364115 (row 4), A364116 (main diagonal), A364117 (first subdiagonal).
%Y Cf. A108625, A143007, A364298.
%K nonn,tabl,easy
%O 0,5
%A _Peter Bala_, Jul 07 2023