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

%I A247853 #24 Feb 16 2025 08:33:23
%S A247853 1680,-1648,-880,36240,5324432,52065680,279702672,1085747600,
%T A247853 3409634960,9202368912,22149057680,48721749392,99650305680,
%U A247853 191909371280,351329819792,615953377680,1040259450512,1700403497360,2700616621200,4180926365072,6326369025680
%N A247853 The 8th Hermite Polynomial evaluated at n: H_8(n) = 256*n^8-3584*n^6+13440*n^4-13440*n^2+1680.
%H A247853 Vincenzo Librandi, <a href="/A247853/b247853.txt">Table of n, a(n) for n = 0..1000</a>
%H A247853 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HermitePolynomial.html">Hermite Polynomial</a>.
%H A247853 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).
%F A247853 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.
%F A247853 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).
%t A247853 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]
%o A247853 (Magma) [256*n^8-3584*n^6+13440*n^4-13440*n^2+1680: n in [0..30]];
%o A247853 (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]];
%o A247853 (PARI) a(n)=polhermite(8,n) \\ _Charles R Greathouse IV_, Jan 29 2016
%o A247853 (Python)
%o A247853 from sympy import hermite
%o A247853 def A247853(n): return hermite(8,n) # _Chai Wah Wu_, Jan 06 2022
%Y A247853 Cf. similar sequences listed in A247850.
%K A247853 sign,easy
%O A247853 0,1
%A A247853 _Vincenzo Librandi_, Sep 25 2014