A159705 - OEIS

%I #22 Sep 08 2022 08:45:44

%S 1,2,-878,-5284,2312620,23267192,-10152119816,-143434219696,

%T 62392319304592,1136856492784160,-492996517654282976,

%U -11013067301664857152,4761026079678523718848,126084356480177895534464,-54337756316633597169242240,-1665565146450503848398045952

%N Numerator of Hermite(n, 1/21).

%H Vincenzo Librandi, <a href="/A159705/b159705.txt">Table of n, a(n) for n = 0..200</a>

%H DLMF <a href="https://dlmf.nist.gov/18.9">Digital library of mathematical functions</a>, Table 18.9.1 for H_n(x)

%F D-finite with recurrence a(n) - 2*a(n-1) + 882*(n-1)*a(n-2) = 0. [DLMF] - _R. J. Mathar_, Feb 17 2014

%F From _G. C. Greubel_, May 22 2018: (Start)

%F a(n) = 21^n * Hermite(n,1/21).

%F E.g.f.: exp(2*x-441*x^2).

%F a(n) = Sum_{k=0..floor(n/2)} (-1)^k*n!*(2/21)^(n-2k)/(k!*(n-2k)!). (End)

%e Numerator of 1, 2/21, -878/441, -5284/9261, 2312620/194481, 23267192/4084101, ...

%p A159705 := proc(n)

%p         orthopoly[H](n,1/21) ;

%p         numer(%) ;

%p end proc: # _R. J. Mathar_, Feb 17 2014

%t Numerator[Table[HermiteH[n, 1/21], {n, 0, 30}]] (* _Vladimir Joseph Stephan Orlovsky_, Jun 17 2011 *)

%o (PARI) a(n)=numerator(polhermite(n,1/21)) \\ _Charles R Greathouse IV_, Jan 29 2016

%o (Magma) [Numerator((&+[(-1)^k*Factorial(n)*(2/21)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // _G. C. Greubel_, May 21 2018

%Y Cf. A009965 (denominators).

%K sign,frac

%O 0,2

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