A364117 a(n) = [x^n] 1/(1 - x) * Legendre_P(n, (1 + x)/(1 - x))^(n+1) for n >= 0.

1, 5, 163, 14409, 2511251, 730485013, 320259339415, 197591579213969, 163325387776051459, 174310058440646865021, 233402385203650889753429, 383208210107883180333696265, 757120215942256247847040802463, 1772210276849283299764079883683173

Offset

0,2

Comments

First subdiagonal of A364113.

Links

Table of n, a(n) for n=0..13.

Formula

Conjectures:

1) the supercongruences a(p) == 2*p + 3 (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.

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);

Crossrefs

Cf. A364113, A364116.

Sequence in context: A195951 A128068 A197095 * A247468 A304056 A305450

Adjacent sequences: A364114 A364115 A364116 * A364118 A364119 A364120

Keyword

nonn,easy

Author

Peter Bala, Jul 08 2023

Status

approved