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%I #25 Sep 08 2022 08:45:43
%S 1,2,-14,-100,556,8312,-33416,-964528,2281360,143454752,-82670816,
%T -25987196992,-35605572416,5542023405440,19415750756224,
%U -1357758396658432,-7957769497497344,375118879242633728,3185315224719454720,-115167886425174418432,-1319713579704402351104
%N Numerator of Hermite(n, 1/3).
%H Vincenzo Librandi, <a href="/A158811/b158811.txt">Table of n, a(n) for n = 0..200</a>
%H DLMF <a href="https://dlmf.nist.gov/18.9">Digital library of mathematical functions</a>, Table 18.9.1 for H_n(x)
%F D-finite with recurrence a(n) -2*a(n-1) +18*(n-1)*a(n-2)=0. [DLMF] - _R. J. Mathar_, Feb 16 2014
%F From _G. C. Greubel_, Jun 09 2018: (Start)
%F a(n) = 3^n * Hermite(n,1/3).
%F E.g.f.: exp(2*x-9*x^2).
%F a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(2/3)^(n-2*k)/(k!*(n-2*k)!)). (End)
%e Numerator of 1, 2/3, -14/9, -100/27, 556/81, 8312/243, -33416/729, -964528/2187, 2281360/6561, 143454752/19683, -82670816/59049,...
%p A158811 := proc(n)
%p orthopoly[H](n,1/3) ;
%p numer(%) ;
%p end proc: # _R. J. Mathar_, Feb 16 2014
%t Numerator[Table[HermiteH[n,1/3],{n,0,50}]] (* _Vladimir Joseph Stephan Orlovsky_, Mar 23 2011 *)
%o (PARI) a(n)=numerator(polhermite(n,1/3)) \\ _Charles R Greathouse IV_, Jan 29 2016
%o (Magma) [Numerator((&+[(-1)^k*Factorial(n)*(2/3)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // _G. C. Greubel_, Jun 09 2018
%Y Cf. A000244 (denominators).
%K sign,frac
%O 0,2
%A _N. J. A. Sloane_, Nov 12 2009
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