A163322 The 3rd Hermite Polynomial evaluated at n: H_3(n) = 8*n^3 - 12*n.
0, -4, 40, 180, 464, 940, 1656, 2660, 4000, 5724, 7880, 10516, 13680, 17420, 21784, 26820, 32576, 39100, 46440, 54644, 63760, 73836, 84920, 97060, 110304, 124700, 140296, 157140, 175280, 194764, 215640, 237956, 261760, 287100, 314024, 342580
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Index entries for sequences related to Hermite polynomials
- Eric Weisstein's World of Mathematics, Hermite Polynomial.
- Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
Programs
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Magma
[8*n^3-12*n: n in [0..40]]; // Vincenzo Librandi, Mar 05 2012
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Maple
A163322 := proc(n) orthopoly[H](3,n) ; end: seq(A163322(n),n=0..80) ; # R. J. Mathar, Jul 26 2009
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Mathematica
CoefficientList[Series[-4*x*(1-14*x+x^2)/(x-1)^4,{x,0,40}],x] (* Vincenzo Librandi, Mar 05 2012 *) LinearRecurrence[{4,-6,4,-1},{0,-4,40,180},40] (* Harvey P. Dale, Aug 14 2014 *) -
PARI
a(n)=8*n^3-12*n \\ Charles R Greathouse IV, Jan 29 2016
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Python
from sympy import hermite def A163322(n): return hermite(3,n) # Chai Wah Wu, Jan 06 2022
Formula
a(n) = 8*n^3 - 12*n.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: -4*x*(1-14*x+x^2)/(x-1)^4.
Extensions
Edited by R. J. Mathar, Jul 26 2009