A364115 a(n) = [x^n] 1/(1 - x) * Legendre_P(n, (1 + x)/(1 - x))^4 for n >= 0.

1, 9, 289, 14409, 908001, 65898009, 5246665201, 445752724041, 39731504675041, 3674479246416009, 349918540195094289, 34125049533650776281, 3394306634561379583281, 343284252364774351717641, 35215197976859176290014289, 3657148830889736882170190409

Offset: 0

Author

Peter Bala, Jul 08 2023

Keywords

• nonn
• easy

Comments

Row 4 of A364113.

Compare with the two types of Apéry numbers A005258 and A005259, which are related to the Legendre polynomials by A005258(n) = [x^n] 1/(1 - x) * Legendre_P(n, (1 + x)/(1 - x)) and A005259(n) = [x^k] 1/(1 - x) * Legendre_P(n, (1 + x)/(1 - x))^2.

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 the present sequence also satisfies the supercongruences 1) and 2).

Examples

			Examples of supercongruences:
a(11) - a(1) = 34125049533650776281 - 9  = (2^4)*(3^2)*(11^3)*13*97*11423* 12360541 == 0 (mod 11^3).
a(11 - 1) - a(0) = 349918540195094289 - 1 = (2^4)*(11^3)*103*159526079101 == 0 (mod 11^3).
a(5^2) - a(5) = 823068999686576893970482230168234294266351898009 - 65898009 = (2^7)*(3^2)*(5^6)*11*17*31*311*35978539*2371705409*297232149579326831 == 0 (mod 5^6).
a(5^2 - 1) - a(5 - 1) = 7402345246022867712987394168675984358488158001- 908001 = (2^4)*(5^6)*13*29*911*1459*26046751*925152076787*2452153330349 == 0 (mod 5^6).
		

Crossrefs

Cf. A005258, A005259, A364113, A364114, A364116.

Programs

Maple

a(n) := coeff(series(1/((1-x))* LegendreP(k,(1+x)/(1-x))^4,x, 21):
seq(a(n), n = 0..20);

Mathematica

Table[SeriesCoefficient[1/(1 - x) * LegendreP[n, (1 + x)/(1 - x)]^4, {x,0,n}], {n,0,20}] (* Vaclav Kotesovec, Jul 09 2023 *)

PARI

a(n) = my(x='x+O('x^(n+1))); polcoef((1/(1-x))*pollegendre(n, (1+x)/(1-x))^4, n); \\ Michel Marcus, Jul 12 2023

Formula

a(n) ~ phi^(10*n + 5) / (2^(3/2) * 5^(1/4) * Pi^(5/2) * n^(5/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Jul 09 2023