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Numerator of Hermite(n, 13/31).
1

%I #15 Sep 08 2022 08:45:45

%S 1,26,-1246,-132340,3743596,1114763416,-6992108744,-13037246540656,

%T -244896579015280,194093391754729376,9282649209429277216,

%U -3489126110080737399104,-286971048447852951277376,73011957343257950639722880,9068569507442760557249311616

%N Numerator of Hermite(n, 13/31).

%H G. C. Greubel, <a href="/A160311/b160311.txt">Table of n, a(n) for n = 0..368</a>

%F From _G. C. Greubel_, Oct 04 2018: (Start)

%F a(n) = 31^n * Hermite(n, 13/31).

%F a(n+2) = 26*a(n+1) - 1922*(n+1)*a(n)

%F E.g.f.: exp(26*x - 961*x^2).

%F a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(26/31)^(n-2*k)/(k!*(n-2*k)!)). (End)

%e Numerators of 1, 26/31, -1246/961, -132340/29791, 3743596/923521, ...

%t Numerator[HermiteH[Range[0,20],13/31]] (* _Harvey P. Dale_, Jan 10 2015 *)

%t Table[31^n*HermiteH[n, 13/31], {n, 0, 30}] (* _G. C. Greubel_, Oct 04 2018 *)

%o (PARI) a(n)=numerator(polhermite(n, 13/31)) \\ _Charles R Greathouse IV_, Jan 29 2016

%o (PARI) x='x+O('x^30); Vec(serlaplace(exp(26*x - 961*x^2))) \\ _G. C. Greubel_, Oct 04 2018

%o (Magma) [Numerator((&+[(-1)^k*Factorial(n)*(26/31)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // _G. C. Greubel_, Oct 04 2018

%Y Cf. A009975 (denominators).

%K sign,frac

%O 0,2

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