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%I #21 Sep 08 2022 08:46:09
%S -30240,8224,200416,-3093984,125984224,3275529760,30232000224,
%T 174454774816,753610971616,2661760648224,8093278209760,21900944840224,
%U 53969864949216,123139662877216,263393520320224,533257736009760,1029623343008224,1907515621443616
%N The 10th Hermite Polynomial evaluated at n: H_10(n) = 1024*n^10 - 23040*n^8 + 161280*n^6 - 403200*n^4 + 302400*n^2 - 30240.
%H Vincenzo Librandi, <a href="/A247855/b247855.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_11">Index entries for linear recurrences with constant coefficients</a>, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1)
%F G.f.: (-30240+340864*x-1553248*x^2+143360*x^3+159704768*x^4 +1703150336*x^5+1689179456*x^6+169683968*x^7-4846240*x^8 +109952*x^9+8224*x^10)/(1-x)^11.
%F a(n) = 11*a(n-1)-55*a(n-2)+165*a(n-3)-330*a(n-4)+462*a(n-5)-462*a(n-6)+330*a(n-7)-165*a(n-8)+55*a(n-9)-11*a(n-10)+a(n-11).
%t Table[1024 n^10 - 23040 n^8 + 161280 n^6 - 403200 n^4 + 302400 n^2 - 30240, {n, 0, 30}] (* or *) CoefficientList[Series[(-30240 + 340864 x - 1553248 x^2 + 143360 x^3 + 159704768 x^4 + 1703150336 x^5 + 1689179456 x^6 + 169683968 x^7 - 4846240 x^8 + 109952 x^9 + 8224 x^10)/(1-x)^11, {x, 0, 30}], x]
%o (Magma) [1024*n^10-23040*n^8+161280*n^6-403200*n^4 +302400*n^2-30240: n in [0..30]]; /* or */ I:=[-30240, 8224,200416,-3093984,125984224,3275529760,30232000224, 174454774816,753610971616,2661760648224,8093278209760]; [n le 11 select I[n] else 11*Self(n-1)-55*Self(n-2)+165*Self(n-3)-330*Self(n-4)+462*Self(n-5)-462*Self(n-6)+330*Self(n-7)-165*Self(n-8)+55*Self(n-9)-11*Self(n-10)+Self(n-11): n in [1..30]];
%o (Python)
%o m = [3715891200, -16721510400, 30036787200, -27095040000, 12533391360, -2612736000, 139499520, 3640320, 153728, -38464, -30240]
%o A247855_list = [m[-1]]
%o for _ in range(10**1):
%o ....for i in range(10):
%o ........m[i+1]+= m[i]
%o ....A247855_list.append(m[-1]) # _Chai Wah Wu_, Nov 09 2014
%o (PARI) a(n)=polhermite(10,n) \\ _Charles R Greathouse IV_, Jan 29 2016
%Y Cf. similar sequences listed in A247850.
%K sign,easy
%O 0,1
%A _Vincenzo Librandi_, Sep 26 2014