A160068 - OEIS

%I #34 Sep 08 2022 08:45:44

%S 1,48,1054,-69408,-7284084,-2596032,45400915464,2198714182272,

%T -291719729560944,-35989688841645312,1554341893161645024,

%U 524479521392325361152,3802815995858998255296,-7684657653083648501025792,-430659327280723849697798016

%N Numerator of Hermite(n, 24/25).

%H Robert Israel, <a href="/A160068/b160068.txt">Table of n, a(n) for n = 0..380</a>

%F From _Robert Israel_, Jan 02 2017 and Jan 24 2017: (Start)

%F Theorem: E.g.f.: exp(-x*(625*x-48)); a(n+2) = -1250*(n+1)*a(n)+48*a(n+1).

%F Proof:

%F The Hermite polynomials H(n,t) have e.g.f. g(x) = exp(-x^2 + 2 x t).

%F Thus b(n) := H(n, 24/25)*5^(2n) has e.g.f. exp(-x*(625*x -48)) and satisfies the recurrence b(n+2) = -1250*(n+1)*b(n)+48*b(n+1), n>=0.

%F To show that these are the numerators of H(n,24/25), we need to show that b(n) is never divisible by 5.  But taking the recurrence mod 5 we get b(n+2) == 3*b(n+1) mod 5.

%F Since b(0) and b(1) are not divisible by 5, induction finishes the proof.

%F (End)

%F From _G. C. Greubel_, Jun 02 2018: (Start)

%F a(n) = 25^n * Hermite(n, 24/25).

%F a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(48/25)^(n-2*k)/(k!*(n-2*k)!)). (End)

%e Numerators of 1, 48/25, 1054/625, -69408/15625, -7284084/390625, ...

%p A160068:=n->numer(orthopoly[H](n,24/25)): seq(A160068(n), n=0..20); # _Wesley Ivan Hurt_, Nov 20 2014

%t Table[Numerator@ HermiteH[n, 24/25], {n, 0, 30}] (* _Michael De Vlieger_, Jan 02 2017 *)

%t Table[25^n*HermiteH[n, 24/25], {n,0,30}] (* _G. C. Greubel_, Jul 11 2018 *)

%o (PARI) a(n)=numerator(polhermite(n,24/25)) \\ _Charles R Greathouse IV_, Jan 29 2016

%o (Magma) [Numerator((&+[(-1)^k*Factorial(n)*(48/25)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // _G. C. Greubel_, Jul 11 2018

%Y Cf. A009969 (denominators).

%K sign,frac

%O 0,2

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