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Numerator of Hermite(n, 3/22).
1

%I #39 Mar 18 2024 12:03:12

%S 1,3,-233,-2151,162705,2570283,-189162201,-4299537519,307542155937,

%T 9246531104595,-642087222317001,-24302866940070903,

%U 1636327584987643953,75484508348928834171,-4921433057324341373625,-270505813458143914292223,17053284557712927443382081

%N Numerator of Hermite(n, 3/22).

%H G. C. Greubel, <a href="/A159807/b159807.txt">Table of n, a(n) for n = 0..435</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 From _Vincenzo Librandi_, Jan 20 2017: (Start)

%F Conjecture: E.g.f.: exp(-x*(121*x-3)).

%F D-finite with recurrence a(n) = 3*a(n-1) - 242*(n-1)*a(n-2). [DLMF] Corrected Feb 06 2021 (End)

%e Numerators of 1, 3/11, -233/121, -2151/1331, 162705/14641, ...

%t Numerator/@HermiteH[Range[0,20],3/22] (* _Harvey P. Dale_, May 01 2011 *)

%o (PARI) a(n)=numerator(polhermite(n, 3/22)) \\ _Charles R Greathouse IV_, Jan 29 2016

%o (SageMath) [numerator(hermite(n, 3/22)) for n in range(20)] # _Bruno Berselli_, Jan 19 2017

%o (Maxima) makelist(num(hermite(n, 3/22)), n, 0, 20); /* _Bruno Berselli_, Jan 19 2017 */

%o (Magma) [Numerator((&+[(-1)^k*Factorial(n)*(3/11)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // _G. C. Greubel_, May 21 2018

%Y Cf. A001020 (denominators).

%K sign,frac

%O 0,2

%A _N. J. A. Sloane_, Nov 12 2009