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Numerator of Hermite(n, 5/28).
2

%I #22 Sep 08 2022 08:45:44

%S 1,5,-367,-5755,402817,11037925,-734331695,-29632858075,1866841880705,

%T 102262852326725,-6074903893493615,-431244900588230075,

%U 24038761085803317505,2148769817796050860325,-111757677404273451703855,-12351237147086094379982875,595378957401697424118753025

%N Numerator of Hermite(n, 5/28).

%H G. C. Greubel, <a href="/A160193/b160193.txt">Table of n, a(n) for n = 0..417</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 D-finite with recurrence a(n) -5*a(n-1) +392*(n-1)*a(n-2)=0. [DLMF] - _R. J. Mathar_, Feb 16 2014

%F From _G. C. Greubel_, Jul 09 2018: (Start)

%F a(n) = 14^n * Hermite(n, 5/28).

%F E.g.f.: exp(5*x - 196*x^2).

%F a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(5/14)^(n-2*k)/(k!*(n-2*k)!)). (End)

%e Numerator of 1, 5/14, -367/196, -5755/2744, 402817/38416, 11037925/537824,..

%p A160193 := proc(n)

%p orthopoly[H](n,5/28) ;

%p numer(%) ;

%p end proc: # _R. J. Mathar_, Feb 16 2014

%t Numerator/@HermiteH[Range[0,20],5/28] (* _Harvey P. Dale_, Jul 11 2011 *)

%t Table[14^n*HermiteH[n, 5/28], {n,0,30}] (* _G. C. Greubel_, Jul 09 2018 *)

%o (PARI) a(n)=numerator(polhermite(n,5/28)) \\ _Charles R Greathouse IV_, Jan 29 2016

%o (Magma) [Numerator((&+[(-1)^k*Factorial(n)*(5/14)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // _G. C. Greubel_, Jul 09 2018

%Y Cf. A001023 (denominators).

%K sign,frac

%O 0,2

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