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

%I #20 Sep 08 2022 08:46:09

%S -120,184,-824,14136,150664,717880,2389704,6412216,14857096,30921144,

%T 59271880,106439224,181253256,295328056,463591624,704861880,

%U 1042468744,1504922296,2126627016,2948642104,4019487880,5395998264,7144219336,9340353976,12071752584

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

%H Vincenzo Librandi, <a href="/A247851/b247851.txt">Table of n, a(n) for n = 0..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HermitePolynomial.html">Hermite Polynomial</a>.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).

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

%F 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).

%t 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]

%t LinearRecurrence[{7,-21,35,-35,21,-7,1},{-120,184,-824,14136,150664,717880,2389704},30] (* _Harvey P. Dale_, Apr 08 2019 *)

%o (Magma) [64*n^6-480*n^4+720*n^2-120: n in [0..30]]; /* or */ 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]]

%o (PARI) a(n)=polhermite(6,n) \\ _Charles R Greathouse IV_, Jan 29 2016

%o (Python)

%o from sympy import hermite

%o def A247851(n): return hermite(6,n) # _Chai Wah Wu_, Jan 06 2022

%Y Cf. similar sequences listed in A247850.

%K sign,easy

%O 0,1

%A _Vincenzo Librandi_, Sep 25 2014