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%I #34 Sep 08 2022 08:45:46
%S 12,-20,76,876,3340,8812,19020,36076,62476,101100,155212,228460,
%T 324876,448876,605260,799212,1036300,1322476,1664076,2067820,2540812,
%U 3090540,3724876,4452076,5280780,6220012,7279180,8468076,9796876,11276140
%N The 4th Hermite Polynomial evaluated at n: H_4(n) = 16n^4 - 48n^2 + 12.
%H Vincenzo Librandi, <a href="/A163323/b163323.txt">Table of n, a(n) for n = 0..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HermitePolynomial.html">Hermite Polynomial</a>
%H <a href="/index/He#Hermite">Index entries for sequences related to Hermite polynomials</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F a(n) = 16*n^4 - 48*n^2 + 12.
%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
%F G.f.: 4*(-3 +20*x -74*x^2 -44*x^3 +5*x^4)/(x-1)^5.
%F H_(m+1)(x) = 2*x*H_m(x) - 2*m*H_(m-1)(x), with H_0(x)=1, H_1(x)=2x.
%p A163323 := proc(n) orthopoly[H](4,n) ; end: seq(A163323(n),n=0..80) ; # _R. J. Mathar_, Jul 26 2009
%t Table[HermiteH[4,n],{n,0,50}] (* _Vladimir Joseph Stephan Orlovsky_, Nov 03 2009 *)
%t Table[16 n^4 - 48 n^2 + 12, {n, 0, 30}] (* _Vincenzo Librandi_, Sep 25 2014 *)
%t LinearRecurrence[{5,-10,10,-5,1},{12,-20,76,876,3340},40] (* _Harvey P. Dale_, Jul 03 2019 *)
%o (Magma) [16*n^4-48*n^2+12: n in [0..40]]; // _Vincenzo Librandi_, Mar 05 2012
%o (PARI) a(n)=16*n^4-48*n^2+12 \\ _Charles R Greathouse IV_, Jan 29 2016
%o (Python)
%o from sympy import hermite
%o def A163323(n): return hermite(4,n) # _Chai Wah Wu_, Jan 06 2022
%Y Cf. A059343, A060821, A163322.
%K sign,easy
%O 0,1
%A _Vincenzo Librandi_, Jul 25 2009
%E Edited by _R. J. Mathar_, Jul 26 2009
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