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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.
6
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, 33001, 11253, 1, 1, 13, 451, 14409, 235251, 819005, 104959, 1, 1, 15, 649, 29531, 908001, 11009257, 21460825, 1004307, 1, 1, 17, 883, 52717, 2511251, 65898009, 554159719, 584307365, 9793891, 1
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
Cf. A005258 (row 1), A005259 (row 2), A364114 (row 3), A364115 (row 4), A364116 (main diagonal), A364117 (first subdiagonal).
Sequence in context: A195522 A273169 A273167 * A324284 A300791 A069972
KEYWORD
nonn,tabl,easy
AUTHOR
Peter Bala, Jul 07 2023
STATUS
approved