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Numerator of Hermite(n, 7/27).
1

%I #15 Sep 08 2022 08:45:44

%S 1,14,-1262,-58492,4701100,406940744,-28573848584,-3959951508688,

%T 236185377526672,49495469682710240,-2406287948347046624,

%U -755331979250773951936,28017398406079098428608,13607531886656648441072768,-340536322975630153440817280

%N Numerator of Hermite(n, 7/27).

%H G. C. Greubel, <a href="/A160107/b160107.txt">Table of n, a(n) for n = 0..376</a>

%F From _G. C. Greubel_, Sep 24 2018: (Start)

%F a(n) = 27^n * Hermite(n, 7/27).

%F E.g.f.: exp(14*x - 729*x^2).

%F a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(14/27)^(n-2*k)/(k!*(n-2*k)!)). (End)

%e Numerators of 1, 14/27, -1262/729, -58492/19683, 4701100/531441, ...

%t HermiteH[Range[0,20],7/27]//Numerator (* _Harvey P. Dale_, Jun 08 2018 *)

%t Table[27^n*HermiteH[n, 7/27], {n, 0, 30}] (* _G. C. Greubel_, Sep 24 2018 *)

%o (PARI) a(n)=numerator(polhermite(n, 7/27)) \\ _Charles R Greathouse IV_, Jan 29 2016

%o (PARI) x='x+O('x^30); Vec(serlaplace(exp(14*x - 729*x^2))) \\ _G. C. Greubel_, Sep 24 2018

%o (Magma) [Numerator((&+[(-1)^k*Factorial(n)*(14/27)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // _G. C. Greubel_, Sep 24 2018

%Y Cf. A009971 (denominators).

%K sign,frac

%O 0,2

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