A364302 a(n) = [x^n] 1/(1 + x) * Legendre_P(n, (1 - x)/(1 + x))^(-n-1) for n >= 0.

1, 3, 163, 23623, 6751251, 3219777011, 2313306332191, 2337707082109071, 3163417897474821763, 5524913023443862515019, 12101947272421487464092429, 32493996621780038121738419591, 104964758754905547830609842389527, 401618040258524641485654323795309235

Offset: 0

Author

Peter Bala, Jul 18 2023

Keywords

• nonn
• easy

Comments

First subdiagonal of A364298.

Crossrefs

Cf. A364117, A364298, A364301.

Programs

Maple

a(n) := coeff(series( 1/(1 + x) * LegendreP(n, (1 - x)/(1 + x))^(-n-1), x, 21), x, n):
seq(a(n), n = 0..20);

Formula

Conjectures:

1) the supercongruences a(p) == 2*p + 1 (mod p^3) hold for all primes p >= 5 (checked up to p = 101).

2) the supercongruences a(p - 1) == 1 (mod p^4) hold for all primes p >= 3 (checked up to p = 101).

3) more generally, the supercongruences a(p^k - 1) == 1 (mod p^(3+k)) may hold for all primes p >= 3 and all k >= 1.