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

0,1

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).

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.

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 *)

Prog

(Magma) [16*n^4-48*n^2+12: n in [0..40]]; // Vincenzo Librandi, Mar 05 2012
(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

Crossrefs

Cf. A059343, A060821, A163322.

Sequence in context: A341361 A181701 A025104 * A060159 A167351 A231400

Adjacent sequences: A163320 A163321 A163322 * A163324 A163325 A163326

Keyword

sign,easy

Author

Vincenzo Librandi, Jul 25 2009

Extensions

Edited by R. J. Mathar, Jul 26 2009

Status

approved