A159507 - OEIS

%I #24 Sep 08 2022 08:45:43

%S 1,1,-97,-293,28225,143081,-13687169,-97818797,9291579137,85981515985,

%T -8109191282849,-92371076948149,8649337125963073,117277723616986297,

%U -10901977774859968705,-171807014577365168189,15854100314466788828161,285247499171775372548513

%N Numerator of Hermite(n, 1/14).

%H T. D. Noe, <a href="/A159507/b159507.txt">Table of n, a(n) for n = 0..100</a>

%F a(n) = Sum_{k = 0..n/2} (-49)^k * n! / (k! * (n - 2*k)!). - _Michael Somos_, Jan 24 2014

%F 0 = a(n) * (-98*a(n+1) + a(n+2) - a(n+3)) + a(n+1) * (-a(n+1) + a(n+2)) for all n in Z. - _Michael Somos_, Jan 24 2014

%F From _G. C. Greubel_, Jun 09 2018: (Start)

%F a(n) = 7^n * Hermite(n,1/14).

%F E.g.f.: exp(x-49*x^2).

%F a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(1/7)^(n-2*k)/(k!*(n-2*k)!)). (End)

%e G.f. = 1 + x - 97*x^2 - 293*x^3 + 28225*x^4 + 143081*x^5 - 13687169*x^6 + ...

%t Numerator[Table[HermiteH[n, 1/14], {n, 0, 50}]] (* _Vladimir Joseph Stephan Orlovsky_, Apr 14 2011 *)

%t a[ n_] := If[ n < 0, 0, HermiteH[n, 1/14] 7^n]; (* _Michael Somos_, Jan 24 2014 *)

%t a[ n_] := Sum[(-49)^k n! / (k! (n - 2 k)!), {k, 0, n/2}]; (* _Michael Somos_, Jan 24 2014 *)

%o (PARI) {a(n) = if( n<0, 0, sum(k=0, n\2, (-49)^k * n! / (k! * (n - 2*k)!)))}; \\ _Michael Somos_, Jan 24 2014

%o (Magma) [Numerator((&+[(-1)^k*Factorial(n)*(1/7)^(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

%Y Cf. A159280, A159488.

%K sign,frac

%O 0,3

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