A158967 Numerator of Hermite(n, 4/5).
1, 8, 14, -688, -7604, 76768, 2515144, -2909248, -903574384, -6064895872, 358089305824, 5897162382592, -149771819142464, -4736471982694912, 59459906581042304, 3791209640534776832, -14265252811503513344, -3147089734919849572352
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..450
- DLMF, Digital library of mathematical functions, Table 18.9.1 for H_n(x).
Programs
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Magma
[Numerator((&+[(-1)^k*Factorial(n)*(8/5)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Jul 13 2018
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Mathematica
Numerator[Table[HermiteH[n,4/5],{n,0,50}]] (* Vladimir Joseph Stephan Orlovsky, Apr 01 2011*) Table[5^n*HermiteH[n, 4/5], {n,0,30}] (* G. C. Greubel, Jul 14 2018 *) -
PARI
a(n)=numerator(polhermite(n,4/5)) \\ Charles R Greathouse IV, Jan 29 2016
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PARI
x='x+O('x^30); Vec(serlaplace(exp(8*x - 25*x^2))) \\ G. C. Greubel, Jul 14 2018
Formula
From G. C. Greubel, Jul 14 2018: (Start)
a(n) = 5^n * Hermite(n, 4/5).
E.g.f.: exp(8*x - 25*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(8/5)^(n-2*k)/(k!*(n-2*k)!)). (End)
D-finite with recurrence a(n) -8*a(n-1) +50*(n-1)*a(n-2)=0. - [DLMF] Georg Fischer, Feb 06 2021