A160088 - OEIS

%I #13 Sep 08 2022 08:45:44

%S 1,4,-1442,-17432,6237580,126613744,-44965503224,-1287479045408,

%T 453768009722512,16832227624528960,-5887014913080686624,

%U -268961938417954983296,93340097422316232142528,5079118464249805316316928,-1748851732685582642764208000

%N Numerator of Hermite(n, 2/27).

%H G. C. Greubel, <a href="/A160088/b160088.txt">Table of n, a(n) for n = 0..375</a>

%F From _G. C. Greubel_, Sep 23 2018: (Start)

%F a(n) = 27^n * Hermite(n, 2/27).

%F E.g.f.: exp(4*x - 729*x^2).

%F a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(4/27)^(n-2*k)/(k!*(n-2*k)!)). (End)

%e Numerators of 1, 4/27, -1442/729, -17432/19683, 6237580/531441...

%t Numerator[HermiteH[Range[0,20],2/27]] (* _Harvey P. Dale_, Mar 26 2016 *)

%t Table[27^n*HermiteH[n, 2/27], {n, 0, 30}] (* _G. C. Greubel_, Sep 23 2018 *)

%o (PARI) a(n)=numerator(polhermite(n, 2/27)) \\ _Charles R Greathouse IV_, Jan 29 2016

%o (PARI) x='x+O('x^30); Vec(serlaplace(exp(4*x - 729*x^2))) \\ _G. C. Greubel_, Sep 23 2018

%o (Magma) [Numerator((&+[(-1)^k*Factorial(n)*(4/27)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // _G. C. Greubel_, Sep 23 2018

%Y Cf. A009971 (denominators).

%K sign,frac

%O 0,2

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