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A025165 a(n) = H_n(1) / 2^floor(n/2) where H_n is the n-th Hermite polynomial.

%I A025165 #21 Sep 08 2022 08:44:48
%S A025165 1,2,1,-2,-5,-2,23,58,-103,-670,257,7214,4387,-77794,-134825,819466,
%T A025165 2841841,-7427774,-55739071,22221790,1081264139,1718092478,
%U A025165 -20988454441,-79774943398,402959508745
%N A025165 a(n) = H_n(1) / 2^floor(n/2) where H_n is the n-th Hermite polynomial.
%H A025165 G. C. Greubel, <a href="/A025165/b025165.txt">Table of n, a(n) for n = 0..800</a>
%H A025165 <a href="/index/He#Hermite">Index entries for sequences related to Hermite polynomials</a>
%F A025165 a(n) = A062267(n)/A016116(n). - _R. J. Mathar_, Feb 05 2013
%F A025165 Conjecture: a(n) +a(n-1) +(2*n-5)*a(n-2) +(2*n-7)*a(n-3) +(n-2)*(n-3)*a(n-4) +(n-3)*(n-4)*a(n-5)=0. - _R. J. Mathar_, Feb 25 2015
%p A025165 A025165 := proc(n)
%p A025165     HermiteH(n,1)/2^(floor(n/2)) ;
%p A025165     simplify(%) ;
%p A025165 end proc: # _R. J. Mathar_, Feb 05 2013
%t A025165 Table[ HermiteH[ n, 1 ]/2^Floor[ n/2 ], {n, 0, 24} ]
%o A025165 (PARI) for(n=0,30, print1(polhermite(n,1)/2^(floor(n/2)), ", ")) \\ _G. C. Greubel_, Jul 10 2018
%o A025165 (Magma) [((&+[(-1)^k*Factorial(n)*(2)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]]))/2^(Floor(n/2)): n in [0..30]]; // _G. C. Greubel_, Jul 10 2018
%K A025165 sign
%O A025165 0,2
%A A025165 _Wouter Meeussen_