A247853 The 8th Hermite Polynomial evaluated at n: H_8(n) = 256*n^8-3584*n^6+13440*n^4-13440*n^2+1680.

1680, -1648, -880, 36240, 5324432, 52065680, 279702672, 1085747600, 3409634960, 9202368912, 22149057680, 48721749392, 99650305680, 191909371280, 351329819792, 615953377680, 1040259450512, 1700403497360, 2700616621200, 4180926365072, 6326369025680

Offset: 0

Author

Vincenzo Librandi, Sep 25 2014

Keywords

• sign
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Eric Weisstein's World of Mathematics, Hermite Polynomial.
• Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Crossrefs

Cf. similar sequences listed in A247850.

Programs

Magma

[256*n^8-3584*n^6+13440*n^4-13440*n^2+1680: n in [0..30]];

Magma

I:=[1680,-1648,-880,36240,5324432, 52065680,279702672,1085747600,3409634960]; [n le 9 select I[n] else 9*Self(n-1)-36*Self(n-2)+84*Self(n-3)-126*Self(n-4)+126*Self(n-5)-84*Self(n-6)+36*Self(n-7)-9*Self(n-8)+Self(n-9): n in [1..30]];

Mathematica

Table[256 n^8 - 3584 n^6 + 13440 n^4 - 13440 n^2 + 1680, {n, 0, 30}] (* or *) CoefficientList[Series[(1680 - 16768 x + 74432 x^2 - 156288 x^3 + 5316704 x^4 + 5105024 x^5 - 15168 x^6 + 13952 x^7 - 1648 x^8)/(1 - x)^9, {x, 0, 30}],x]

PARI

a(n)=polhermite(8,n) \\ Charles R Greathouse IV, Jan 29 2016

Python

from sympy import hermite
def A247853(n): return hermite(8,n) # Chai Wah Wu, Jan 06 2022

Formula

G.f.: (1680-16768*x+74432*x^2-156288*x^3+5316704*x^4 +5105024*x^5-15168*x^6+13952*x^7-1648*x^8)/(1-x)^9.

a(n) = 9*a(n-1)-36*a(n-2)+84*a(n-3)-126*a(n-4)+126*a(n-5)-84*a(n-6)+36*a(n-7)-9*a(n-8)+a(n-9).