|
%I #20 Sep 08 2022 08:45:44
%S 1,8,-818,-20656,1999180,88867808,-8105441336,-535131970624,
%T 45761939043472,4141986697070720,-330122378550514976,
%U -39173301696567870208,2889460903124553335488,437725912381470764965376,-29628751416174362424982400,-5642069577415795905192322048
%N Numerator of Hermite(n, 4/21).
%H Vincenzo Librandi, <a href="/A159707/b159707.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) - 8*a(n-1) + 882*(n-1)*a(n-2) = 0. [DLMF] - _R. J. Mathar_, Feb 17 2014
%F From _G. C. Greubel_, May 22 2018: (Start)
%F a(n) = 21^n * Hermite(n,4/21).
%F E.g.f.: exp(8*x-441*x^2).
%F a(n) = Sum_{k=0..floor(n/2)} (-1)^k*n!*(8/21)^(n-2k)/(k!*(n-2k)!). (End)
%e Numerator of 1, 8/21, -818/441, -20656/9261, 1999180/194481, 88867808/4084101, ...
%p A159707 := proc(n)
%p orthopoly[H](n,4/21) ;
%p numer(%) ;
%p end proc: # _R. J. Mathar_, Feb 17 2014
%t Numerator[Table[HermiteH[n, 4/21], {n, 0, 30}]] (* _Vladimir Joseph Stephan Orlovsky_, Jun 17 2011 *)
%o (PARI) a(n)=numerator(polhermite(n,4/21)) \\ _Charles R Greathouse IV_, Jan 29 2016
%o (Magma) [Numerator((&+[(-1)^k*Factorial(n)*(8/21)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // _G. C. Greubel_, May 22 2018
%Y Cf. A009965 (denominators).
%K sign,frac
%O 0,2
%A _N. J. A. Sloane_, Nov 12 2009
| 