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

%I #21 Sep 08 2022 08:44:48

%S 1,2,1,-2,-5,-2,23,58,-103,-670,257,7214,4387,-77794,-134825,819466,

%T 2841841,-7427774,-55739071,22221790,1081264139,1718092478,

%U -20988454441,-79774943398,402959508745

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

%H G. C. Greubel, <a href="/A025165/b025165.txt">Table of n, a(n) for n = 0..800</a>

%H <a href="/index/He#Hermite">Index entries for sequences related to Hermite polynomials</a>

%F a(n) = A062267(n)/A016116(n). - _R. J. Mathar_, Feb 05 2013

%F 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 := proc(n)

%p HermiteH(n,1)/2^(floor(n/2)) ;

%p simplify(%) ;

%p end proc: # _R. J. Mathar_, Feb 05 2013

%t Table[ HermiteH[ n, 1 ]/2^Floor[ n/2 ], {n, 0, 24} ]

%o (PARI) for(n=0,30, print1(polhermite(n,1)/2^(floor(n/2)), ", ")) \\ _G. C. Greubel_, Jul 10 2018

%o (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 sign

%O 0,2

%A _Wouter Meeussen_