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Numerator of Hermite(n, 3/7).
2

%I #25 Sep 08 2022 08:45:43

%S 1,6,-62,-1548,8940,660456,-417864,-390855312,-2058477168,

%T 294079701600,3580055071776,-266717777137344,-5459606030198592,

%U 280902469732324992,8640952900866956160,-333552471067548152064,-14703515590679714467584,434789181089837215630848

%N Numerator of Hermite(n, 3/7).

%H Vincenzo Librandi, <a href="/A158987/b158987.txt">Table of n, a(n) for n = 0..200</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) - 6*a(n-1) + 98*(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) = 7^n * Hermite(n, 3/7).

%F E.g.f.: exp(6*x-49*x^2).

%F a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(6/7)^(n-2*k)/(k!*(n-2*k)!)). (End)

%e Numerator of 1, 6/7, -62/49, -1548/343, 8940/2401, 660456/16807, -417864/117649, ...

%p A158987 := proc(n)

%p orthopoly[H](n,3/7) ;

%p numer(%) ;

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

%t Numerator[Table[HermiteH[n,3/7],{n,0,50}]] (* _Vladimir Joseph Stephan Orlovsky_, Mar 23 2011 *)

%t Table[7^n*HermiteH[n, 3/7], {n,0,30}] (* _G. C. Greubel_, Jul 09 2018 *)

%o (PARI) a(n)=numerator(polhermite(n,3/7)) \\ _Charles R Greathouse IV_, Jan 29 2016

%o (Magma) [Numerator((&+[(-1)^k*Factorial(n)*(6/7)^(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. A000420 (denominators).

%K sign,frac

%O 0,2

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