OFFSET
0,5
COMMENTS
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.
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.
EXAMPLE
Square array begins
n\k| 0 1 2 3 4 5 6 7
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
0 | 1 1 1 1 1 1 1 1
1 | 1 3 19 147 1251 11253 104959 1004307
2 | 1 5 73 1445 33001 819005 21460825 584307365
3 | 1 7 163 5623 235251 11009257 554159719 29359663991
4 | 1 9 289 14409 908001 65898009 5246665201 445752724041
5 | 1 11 451 29531 2511251 251831261 28224521263 3423024241627
6 | 1 13 649 52717 5665001 730485013 106898093065 17144295476461
MAPLE
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 = 0..10);
# display as a sequence
seq(seq(T(n-k, k), k = 0..n), n = 0..10);
CROSSREFS
KEYWORD
AUTHOR
Peter Bala, Jul 07 2023
STATUS
approved