A247852 The 7th Hermite Polynomial evaluated at n: H_7(n) = 128*n^7 -1344*n^5 + 3360*n^3 - 1680*n.

0, 464, -3104, 39024, 929216, 6211600, 26096544, 83965616, 226102144, 535292496, 1148943200, 2282359024, 4257827136, 7540152464, 12779289376, 20860714800, 32964187904, 50631541456, 75844149984, 111110719856, 159566046400, 225081383184, 312387068576

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 (8,-28,56,-70,56,-28,8,-1).

Crossrefs

Cf. similar sequences listed in A247850.

Programs

Magma

[128*n^7-1344*n^5+3360*n^3-1680*n: n in [0..30]];

Magma

I:=[0,464,-3104,39024,929216,6211600,26096544,83965616]; [n le 8 select I[n] else 8*Self(n-1)-28*Self(n-2)+56*Self(n-3)-70*Self(n-4)+56*Self(n-5)-28*Self(n-6)+8*Self(n-7)-Self(n-8): n in [1..30]];

Mathematica

Table[128 n^7 - 1344 n^5 + 3360 n^3 - 1680 n, {n, 0, 30}] (* or *) CoefficientList[Series[x (464 - 6816 x + 76848 x^2 + 504128 x^3 + 76848 x^4 - 6816 x^5 + 464 x^6)/(1-x)^8, {x, 0, 30}], x]

PARI

a(n)=polhermite(7,n) \\ Charles R Greathouse IV, Jan 29 2016

Python

from sympy import hermite
def A247852(n): return hermite(7, n) # Chai Wah Wu, Jan 06 2022

Formula

G.f.: x*(464-6816*x+76848*x^2+504128*x^3+76848*x^4-6816*x^5 +464*x^6)/(1-x)^8.

a(n) = 8*a(n-1)-28*a(n-2)+56*a(n-3)-70*a(n-4)+56*a(n-5)-28*a(n-6)+8*a(n-7)-a(n-8).