A285199 Product of n! and the n-th Legendre polynomial evaluated at 2.

1, 2, 11, 102, 1329, 22290, 457155, 11083590, 310107105, 9834291810, 348584413275, 13657116176550, 586048630115025, 27335806776328050, 1377091347432275475, 74513480022911679750, 4309990208248262162625, 265383902858519912717250, 17331286029912646125208875

Offset

0,2

Links

Table of n, a(n) for n=0..18.

Formula

a(n) = P_n(2)*n!, where P_n is the n-th Legendre polynomial.

E.g.f.: 1/sqrt(1 - 4*x + x^2). - Ilya Gutkovskiy, May 05 2017

D-finite with recurrence: a(n+2) = (4n+6) a(n+1) - (n+1)^2 a(n). - Robert Israel, May 05 2017

a(n) ~ 3^(-1/4) * (2 + sqrt(3))^(n + 1/2) * n^n / exp(n). - Vaclav Kotesovec, May 06 2017

Maple

seq(orthopoly[P](n, 2)*n!, n=0..30); # Robert Israel, May 05 2017

Mathematica

Table[n!*LegendreP[n, 2], {n, 0, 20}] (* Vaclav Kotesovec, May 06 2017 *)

Prog

(Python)
from sympy import legendre, factorial
def a(n): return legendre(n, 2)*factorial(n)
print([a(n) for n in range(21)]) # Indranil Ghosh, May 05 2017
(PARI) a(n) = n!*pollegendre(n, 2); \\ Michel Marcus, May 06 2017

Crossrefs

Sequence in context: A351622 A024721 A367852 * A339081 A081716 A334240

Adjacent sequences: A285196 A285197 A285198 * A285200 A285201 A285202

Keyword

nonn

Author

Jeffrey Shallit, May 05 2017

Status

approved