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A159858 Numerator of Hermite(n, 1/23).

%I A159858 #28 Sep 08 2022 08:45:44
%S A159858 1,2,-1054,-6340,3332716,33496312,-17563075016,-247760738608,
%T A159858 129576612091280,2356200115760672,-1229116100101646816,
%U A159858 -27386829424951203392,14249679514133063237824,376202545407446604740480,-195237686946571258563550336,-5962787476468241626543090432
%N A159858 Numerator of Hermite(n, 1/23).
%C A159858 Consider any odd integer k. B(n) = k^n Hermite(n,1/k) satisfies the recurrence B(n) = 2*B(n-1) - 2*k^2*(n-1)*B(n-2) with B(0) = 1 and B(1) = 2. In particular, B(n) == 2*B(n-1) mod k, and B(n) is coprime to k. Therefore B(n) is the numerator of Hermite(n,1/k). - _Robert Israel_, Jun 27 2014
%H A159858 G. C. Greubel, <a href="/A159858/b159858.txt">Table of n, a(n) for n = 0..385</a>
%H A159858 DLMF <a href="https://dlmf.nist.gov/18.9">Digital library of mathematical functions</a>, Table 18.9.1 for H_n(x)
%F A159858 D-finite with recurrence a(n) - 2*a(n-1) + 1058*(n-1)*a(n-2) = 0. [DLMF] - _R. J. Mathar_, Feb 16 2014
%F A159858 a(n) = 23^n * Hermite(n,1/23). This does satisfy the above formula. - _Robert Israel_, Jun 27 2014
%F A159858 From _G. C. Greubel_, Jun 09 2018: (Start)
%F A159858 E.g.f.: exp(2*x-529*x^2).
%F A159858 a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(2/23)^(n-2*k)/(k!*(n-2*k)!)). (End)
%e A159858 Numerator of 1, 2/23, -1054/529, -6340/12167, 3332716/279841, 33496312/6436343, -17563075016/148035889, -247760738608/3404825447, 129576612091280/78310985281...
%p A159858 A159858 := proc(n)
%p A159858         orthopoly[H](n,1/23) ;
%p A159858         numer(%) ;
%p A159858 end proc: # _R. J. Mathar_, Feb 16 2014
%t A159858 Numerator[Table[HermiteH[n, 1/23], {n, 0, 30}]] (* _Vladimir Joseph Stephan Orlovsky_, Jun 22 2011 *)
%o A159858 (PARI) a(n)=numerator(polhermite(n,1/23)) \\ _Charles R Greathouse IV_, Jan 29 2016
%o A159858 (Magma) [Numerator((&+[(-1)^k*Factorial(n)*(2/23)^(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 A159858 Cf. A009967 (denominators).
%K A159858 sign,frac
%O A159858 0,2
%A A159858 _N. J. A. Sloane_, Nov 12 2009