A247851 The 6th Hermite Polynomial evaluated at n: H_6(n) = 64*n^6-480*n^4+720*n^2-120.

-120, 184, -824, 14136, 150664, 717880, 2389704, 6412216, 14857096, 30921144, 59271880, 106439224, 181253256, 295328056, 463591624, 704861880, 1042468744, 1504922296, 2126627016, 2948642104, 4019487880, 5395998264, 7144219336, 9340353976, 12071752584

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 (7,-21,35,-35,21,-7,1).

Crossrefs

Cf. similar sequences listed in A247850.

Programs

Magma

[64*n^6-480*n^4+720*n^2-120: n in [0..30]];

Magma

I:=[-120,184,-824,14136,150664,717880,2389704]; [n le 7 select I[n] else 7*Self(n-1)-21*Self(n-2)+35*Self(n-3)-35*Self(n-4)+21*Self(n-5)-7*Self(n-6)+Self(n-7): n in [1..30]];

Mathematica

Table[64 n^6 - 480 n^4 + 720 n^2 - 120, {n, 0, 30}] (* or *) CoefficientList[Series[(-120 + 1024 x -4632 x^2 + 27968 x^3 + 23768 x^4 - 2112 x^5 + 184 x^6)/(1-x)^7, {x, 0, 30}], x]
LinearRecurrence[{7,-21,35,-35,21,-7,1},{-120,184,-824,14136,150664,717880,2389704},30] (* Harvey P. Dale, Apr 08 2019 *)

PARI

a(n)=polhermite(6,n) \\ Charles R Greathouse IV, Jan 29 2016

Python

from sympy import hermite
def A247851(n): return hermite(6,n) # Chai Wah Wu, Jan 06 2022

Formula

G.f.: (-120 + 1024*x - 4632*x^2 + 27968*x^3 + 23768*x^4 -2112* x^5 + 184*x^6)/(1-x)^7.

a(n) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7).