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A160077 Numerator of Hermite(n, 19/26). 1

%I

%S 1,19,23,-12407,-259055,11852219,662995111,-11439393023,

%T -1785994900063,-3001784367005,5375962583018551,112289320237829369,

%U -17854331799144214607,-794677787068375998197,63353055971140535017415,4964123351859225388799089,-226881650088357230151111359

%N Numerator of Hermite(n, 19/26).

%H Vincenzo Librandi, <a href="/A160077/b160077.txt">Table of n, a(n) for n = 0..100</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) -19*a(n-1) +338*(n-1)*a(n-2)=0. [DLMF] - _R. J. Mathar_, Feb 16 2014

%F E.g.f.: exp(-x*(169*x-19)). The conjecture is a consequence. - _Robert Israel_, Jan 02 2017

%F From _G. C. Greubel_, Jun 02 2018: (Start)

%F a(n) = 13^n * Hermite(n, 19/26).

%F a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(19/13)^(n-2*k)/(k!*(n-2*k)!)). (End)

%e Numerator of 1, 19/13, 23/169, -12407/2197, -259055/28561, 11852219/371293,...

%p A160077 := proc(n)

%p orthopoly[H](n,19/26) ;

%p numer(%) ;

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

%t f[n_] := HermiteH[n, 19/26]*13^n; Array[f, 17, 0] (* _Robert G. Wilson v_, Nov 13 2011 *)

%t HermiteH[Range[0,30],19/26]//Numerator (* _Harvey P. Dale_, Feb 02 2017 *)

%o (PARI) a(n)=numerator(polhermite(n,19/26)) \\ _Charles R Greathouse IV_, Jan 29 2016

%o (MAGMA) [Numerator((&+[(-1)^k*Factorial(n)*(19/13)^(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. A001022 (denominators)

%K sign,frac

%O 0,2

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

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Last modified July 30 16:44 EDT 2021. Contains 346359 sequences. (Running on oeis4.)