A160068 Numerator of Hermite(n, 24/25).

1, 48, 1054, -69408, -7284084, -2596032, 45400915464, 2198714182272, -291719729560944, -35989688841645312, 1554341893161645024, 524479521392325361152, 3802815995858998255296, -7684657653083648501025792, -430659327280723849697798016

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..380

Formula

From Robert Israel, Jan 02 2017 and Jan 24 2017: (Start)

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

Proof:

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

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.

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.

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

(End)

From G. C. Greubel, Jun 02 2018: (Start)

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

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

Example

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

Maple

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

Mathematica

Table[Numerator@ HermiteH[n, 24/25], {n, 0, 30}] (* Michael De Vlieger, Jan 02 2017 *)
Table[25^n*HermiteH[n, 24/25], {n, 0, 30}] (* G. C. Greubel, Jul 11 2018 *)

Prog

(PARI) a(n)=numerator(polhermite(n, 24/25)) \\ Charles R Greathouse IV, Jan 29 2016
(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

Crossrefs

Cf. A009969 (denominators).

Sequence in context: A292045 A341306 A272778 * A229387 A010839 A000156

Adjacent sequences: A160065 A160066 A160067 * A160069 A160070 A160071

Keyword

sign,frac

Author

N. J. A. Sloane, Nov 12 2009

Status

approved