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A364113 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.

%I A364113 #17 Jul 22 2023 21:19:56
%S A364113 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 A364113 33001,11253,1,1,13,451,14409,235251,819005,104959,1,1,15,649,29531,
%U A364113 908001,11009257,21460825,1004307,1,1,17,883,52717,2511251,65898009,554159719,584307365,9793891,1
%N A364113 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 A364113 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 A364113 Both types of Apéry numbers satisfy the supercongruences
%C A364113 1) u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r))
%C A364113 and the shifted supercongruences
%C A364113 2) u(n*p^r - 1) == u(n*p^(r-1) - 1) (mod p^(3*r))
%C A364113 for all primes p >= 5 and positive integers n and r.
%C A364113 We conjecture that each row sequence of the present table satisfies the same pair of supercongruences.
%e A364113 Square array begins
%e A364113  n\k|  0   1    2      3        4          5             6               7
%e A364113   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
%e A364113   0 |  1   1    1      1        1          1             1               1
%e A364113   1 |  1   3   19    147     1251      11253        104959         1004307
%e A364113   2 |  1   5   73   1445    33001     819005      21460825       584307365
%e A364113   3 |  1   7  163   5623   235251   11009257     554159719     29359663991
%e A364113   4 |  1   9  289  14409   908001   65898009    5246665201    445752724041
%e A364113   5 |  1  11  451  29531  2511251  251831261   28224521263   3423024241627
%e A364113   6 |  1  13  649  52717  5665001  730485013  106898093065  17144295476461
%p A364113 T(n,k) := coeff(series(1/(1-x)* LegendreP(k,(1+x)/(1-x))^n, x, 11), x, k):
%p A364113 # display as a square array
%p A364113 seq(print(seq(T(n, k), k = 0..10)), n = 0..10);
%p A364113 # display as a sequence
%p A364113 seq(seq(T(n-k, k), k = 0..n), n = 0..10);
%Y A364113 Cf. A005258 (row 1), A005259 (row 2), A364114 (row 3), A364115 (row 4), A364116 (main diagonal), A364117 (first subdiagonal).
%Y A364113 Cf. A108625, A143007, A364298.
%K A364113 nonn,tabl,easy
%O A364113 0,5
%A A364113 _Peter Bala_, Jul 07 2023