A159858 Numerator of Hermite(n, 1/23).

1, 2, -1054, -6340, 3332716, 33496312, -17563075016, -247760738608, 129576612091280, 2356200115760672, -1229116100101646816, -27386829424951203392, 14249679514133063237824, 376202545407446604740480, -195237686946571258563550336, -5962787476468241626543090432

Offset: 0

Author

N. J. A. Sloane, Nov 12 2009

Keywords

• sign
• frac

Comments

Consider any odd integer k. B(n) = k^n Hermite(n,1/k) satisfies the recurrence B(n) = 2*B(n-1) - 2*k^2*(n-1)*B(n-2) with B(0) = 1 and B(1) = 2. In particular, B(n) == 2*B(n-1) mod k, and B(n) is coprime to k. Therefore B(n) is the numerator of Hermite(n,1/k). - Robert Israel, Jun 27 2014

Examples

			Numerator of 1, 2/23, -1054/529, -6340/12167, 3332716/279841, 33496312/6436343, -17563075016/148035889, -247760738608/3404825447, 129576612091280/78310985281...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..385
• DLMF Digital library of mathematical functions, Table 18.9.1 for H_n(x)

Crossrefs

Cf. A009967 (denominators).

Programs

Magma

[Numerator((&+[(-1)^k*Factorial(n)*(2/23)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Jun 09 2018

Maple

A159858 := proc(n)
        orthopoly[H](n,1/23) ;
        numer(%) ;
end proc: # R. J. Mathar, Feb 16 2014

Mathematica

Numerator[Table[HermiteH[n, 1/23], {n, 0, 30}]] (* Vladimir Joseph Stephan Orlovsky, Jun 22 2011 *)

PARI

a(n)=numerator(polhermite(n,1/23)) \\ Charles R Greathouse IV, Jan 29 2016

Formula

D-finite with recurrence a(n) - 2*a(n-1) + 1058*(n-1)*a(n-2) = 0. [DLMF] - R. J. Mathar, Feb 16 2014

a(n) = 23^n * Hermite(n,1/23). This does satisfy the above formula. - Robert Israel, Jun 27 2014

From G. C. Greubel, Jun 09 2018: (Start)

E.g.f.: exp(2*x-529*x^2).

a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(2/23)^(n-2*k)/(k!*(n-2*k)!)). (End)