A158968 Numerator of Hermite(n, 1/6).

1, 1, -17, -53, 865, 4681, -73169, -578717, 8640577, 91975825, -1307797649, -17863446149, 241080488353, 4099584856537, -52313249418065, -1085408633265389, 13039168709612161, 325636855090044193, -3664348770051277073, -109170689819225595605, 1144036589538311163361

Offset: 0

Author

N. J. A. Sloane, Nov 12 2009

Keywords

• sign
• frac

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).

Crossrefs

Cf. A158811, A158960.

Sequences with e.g.f = exp(x + q*x^2): this sequence (q=-9), A158954 (q=-4), A362177 (q=-3), A362176 (q=-2), A293604 (q=-1), A000012 (q=0), A047974 (q=1), A115329 (q=2), A293720 (q=4).

Programs

Magma

[Numerator((&+[(-1)^k*Factorial(n)*(1/3)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Jul 10 2018

Mathematica

Numerator[Table[HermiteH[n,1/6],{n,0,50}]] (* Vladimir Joseph Stephan Orlovsky, Apr 01 2011 *)
Table[3^n*HermiteH[n, 1/6], {n,0, 50}] (* G. C. Greubel, Jul 10 2018 *)

PARI

a(n)=numerator(polhermite(n,1/6)) \\ Charles R Greathouse IV, Jan 29 2016

SageMath

[3^n*hermite(n, 1/6) for n in range(31)] # G. C. Greubel, Jul 12 2024

Formula

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

a(n) = 3^n * Hermite(n, 1/6).

E.g.f.: exp(x - 9*x^2).

a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(1/3)^(n-2*k)/(k!*(n-2*k)!)). (End)

D-finite with recurrence a(n) -a(n-1) +18*(n-1)*a(n-2)=0. - [DLMF] Georg Fischer, Feb 06 2021