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

0,2

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).

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.

Maple

A163322 := proc(n) orthopoly[H](3, n) ; end: seq(A163322(n), n=0..80) ; # R. J. Mathar, Jul 26 2009

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 *)

Prog

(Magma) [8*n^3-12*n: n in [0..40]]; // Vincenzo Librandi, Mar 05 2012
(PARI) a(n)=8*n^3-12*n \\ Charles R Greathouse IV, Jan 29 2016
(Python)
from sympy import hermite
def A163322(n): return hermite(3, n) # Chai Wah Wu, Jan 06 2022

Crossrefs

Cf. A060821, A059343.

Sequence in context: A248964 A224086 A271013 * A238328 A009355 A355838

Adjacent sequences: A163319 A163320 A163321 * A163323 A163324 A163325

Keyword

sign,easy

Author

Vincenzo Librandi, Jul 25 2009

Extensions

Edited by R. J. Mathar, Jul 26 2009

Status

approved