login
A285199
Product of n! and the n-th Legendre polynomial evaluated at 2.
0
1, 2, 11, 102, 1329, 22290, 457155, 11083590, 310107105, 9834291810, 348584413275, 13657116176550, 586048630115025, 27335806776328050, 1377091347432275475, 74513480022911679750, 4309990208248262162625, 265383902858519912717250, 17331286029912646125208875
OFFSET
0,2
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
KEYWORD
nonn
AUTHOR
Jeffrey Shallit, May 05 2017
STATUS
approved