A285199 - OEIS

%I #40 May 11 2021 08:56:35

%S 1,2,11,102,1329,22290,457155,11083590,310107105,9834291810,

%T 348584413275,13657116176550,586048630115025,27335806776328050,

%U 1377091347432275475,74513480022911679750,4309990208248262162625,265383902858519912717250,17331286029912646125208875

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

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

%F E.g.f.: 1/sqrt(1 - 4*x + x^2). - _Ilya Gutkovskiy_, May 05 2017

%F D-finite with recurrence: a(n+2) = (4n+6) a(n+1) - (n+1)^2 a(n). - _Robert Israel_, May 05 2017

%F a(n) ~ 3^(-1/4) * (2 + sqrt(3))^(n + 1/2) * n^n / exp(n). - _Vaclav Kotesovec_, May 06 2017

%p seq(orthopoly[P](n,2)*n!, n=0..30); # _Robert Israel_, May 05 2017

%t Table[n!*LegendreP[n, 2], {n, 0, 20}] (* _Vaclav Kotesovec_, May 06 2017 *)

%o (Python)

%o from sympy import legendre, factorial

%o def a(n): return legendre(n, 2)*factorial(n)

%o print([a(n) for n in range(21)]) # _Indranil Ghosh_, May 05 2017

%o (PARI) a(n) = n!*pollegendre(n, 2); \\ _Michel Marcus_, May 06 2017

%K nonn

%O 0,2

%A _Jeffrey Shallit_, May 05 2017