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A158954 Numerator of Hermite(n, 1/4).

%I A158954 #33 Jul 13 2024 02:14:38
%S A158954 1,1,-7,-23,145,881,-4919,-47207,228257,3249505,-13184999,-273145399,
%T A158954 887134513,27109092817,-65152896535,-3101371292039,4716976292161,
%U A158954 401692501673153,-239816274060743,-58083536514994775,-21631462857761839,9271734379541402161
%N A158954 Numerator of Hermite(n, 1/4).
%H A158954 Vincenzo Librandi, <a href="/A158954/b158954.txt">Table of n, a(n) for n = 0..200</a>
%H A158954 DLMF <a href="https://dlmf.nist.gov/18.9">Digital library of mathematical functions</a>, Table 18.9.1 for H_n(x)
%F A158954 D-finite with recurrence a(n) - a(n-1) + 8*(n-1)*a(n-2) = 0. [DLMF] - _R. J. Mathar_, Feb 16 2014
%F A158954 From _G. C. Greubel_, Jun 09 2018: (Start)
%F A158954 a(n) = 2^n*Hermite(n,1/4).
%F A158954 E.g.f.: exp(x-4*x^2).
%F A158954 a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(1/2)^(n-2k)/(k!*(n-2k)!)). (End)
%e A158954 Numerators of 1, 1/2, -7/4, -23/8, 145/16, 881/32, -4919/64, -47207/128, 228257/256, 3249505/512, ...
%p A158954 A158954 := proc(n)
%p A158954         orthopoly[H](n,1/4) ;
%p A158954         numer(%) ;
%p A158954 end proc: # _R. J. Mathar_, Feb 16 2014
%t A158954 Numerator[Table[HermiteH[n,1/4],{n,0,50}]] (* _Vladimir Joseph Stephan Orlovsky_, Mar 23 2011 *)
%o A158954 (PARI) a(n)=numerator(polhermite(n,1/4)) \\ _Charles R Greathouse IV_, Jan 29 2016
%o A158954 (Magma) [Numerator((&+[(-1)^k*Factorial(n)*(1/2)^(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
%o A158954 (SageMath) [2^n*hermite(n, 1/4) for n in range(31)] # _G. C. Greubel_, Jul 12 2024
%Y A158954 Cf. A000079 (denominators).
%Y A158954 Sequences with e.g.f = exp(x + q*x^2): A158968 (q=-9), this sequence (q=-4), A362177 (q=-3), A362176 (q=-2), A293604 (q=-1), A000012 (q=0), A047974 (q=1), A115329 (q=2), A293720 (q=4).
%K A158954 sign,frac
%O A158954 0,3
%A A158954 _N. J. A. Sloane_, Nov 12 2009