A163323 The 4th Hermite Polynomial evaluated at n: H_4(n) = 16n^4 - 48n^2 + 12.
12, -20, 76, 876, 3340, 8812, 19020, 36076, 62476, 101100, 155212, 228460, 324876, 448876, 605260, 799212, 1036300, 1322476, 1664076, 2067820, 2540812, 3090540, 3724876, 4452076, 5280780, 6220012, 7279180, 8468076, 9796876, 11276140
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Eric Weisstein's World of Mathematics, Hermite Polynomial
- Index entries for sequences related to Hermite polynomials
- Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
Programs
-
Magma
[16*n^4-48*n^2+12: n in [0..40]]; // Vincenzo Librandi, Mar 05 2012
-
Maple
A163323 := proc(n) orthopoly[H](4,n) ; end: seq(A163323(n),n=0..80) ; # R. J. Mathar, Jul 26 2009
-
Mathematica
Table[HermiteH[4,n],{n,0,50}] (* Vladimir Joseph Stephan Orlovsky, Nov 03 2009 *) Table[16 n^4 - 48 n^2 + 12, {n, 0, 30}] (* Vincenzo Librandi, Sep 25 2014 *) LinearRecurrence[{5,-10,10,-5,1},{12,-20,76,876,3340},40] (* Harvey P. Dale, Jul 03 2019 *) -
PARI
a(n)=16*n^4-48*n^2+12 \\ Charles R Greathouse IV, Jan 29 2016
-
Python
from sympy import hermite def A163323(n): return hermite(4,n) # Chai Wah Wu, Jan 06 2022
Formula
a(n) = 16*n^4 - 48*n^2 + 12.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: 4*(-3 +20*x -74*x^2 -44*x^3 +5*x^4)/(x-1)^5.
H_(m+1)(x) = 2*x*H_m(x) - 2*m*H_(m-1)(x), with H_0(x)=1, H_1(x)=2x.
Extensions
Edited by R. J. Mathar, Jul 26 2009