A159656 - OEIS

%I #20 Sep 08 2022 08:45:44

%S 1,36,574,-31320,-2370804,5103216,8742318216,292616324064,

%T -33649488597360,-2901533477298624,114199171722894816,

%U 25060241888120278656,-4801113850900597056,-217294775817306515769600,-7777548674818481563737984,1916423841667868925104549376

%N Numerator of Hermite(n, 18/19).

%H Vincenzo Librandi, <a href="/A159656/b159656.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) -36*a(n-1) +722*(n-1)*a(n-2)=0. [DLMF] - _R. J. Mathar_, Feb 16 2014

%F From _G. C. Greubel_, Jul 11 2018: (Start)

%F a(n) = 19^n * Hermite(n, 18/19).

%F E.g.f.: exp(36*x - 361*x^2).

%F a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(36/19)^(n-2*k)/(k!*(n-2*k)!)). (End)

%e Numerator of 1, 36/19, 574/361, -31320/6859, -2370804/130321, 5103216/2476099,...

%p A159656 := proc(n)

%p         orthopoly[H](n,18/19) ;

%p         numer(%) ;

%p end proc: # _R. J. Mathar_, Feb 16 2014

%t Numerator[Table[HermiteH[n, 18/19], {n, 0, 30}]] (* _Vladimir Joseph Stephan Orlovsky_, Jun 16 2011 *)

%t Table[19^n*HermiteH[n,18/19], {n,0,50}] (* _G. C. Greubel_, Jul 11 2018 *)

%o (PARI) a(n)=numerator(polhermite(n,18/19)) \\ _Charles R Greathouse IV_, Jan 29 2016

%o (Magma) [Numerator((&+[(-1)^k*Factorial(n)*(36/19)^(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. A001029 (denominators).

%K sign,frac

%O 0,2

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