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.

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.

Links

Table of n, a(n) for n=0..54.

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).

Cf. A108625, A143007, A364298.

Sequence in context: A195522 A273169 A273167 * A324284 A300791 A069972

Adjacent sequences: A364110 A364111 A364112 * A364114 A364115 A364116

Keyword

nonn,tabl,easy

Author

Peter Bala, Jul 07 2023

Status

approved