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%I #33 Sep 08 2022 08:44:47
%S 265,2119,18806,183822,1965624,22852200,287250480,3884393520,
%T 56255149440,869007242880,14266826784000,248112809683200,
%U 4557208289356800,88166812070937600,1792259345728051200,38195370237024000000,851609625265631232000,19827505082582765568000
%N 6th differences of factorial numbers.
%H Vincenzo Librandi, <a href="/A023043/b023043.txt">Table of n, a(n) for n = 0..200</a>
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a>
%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F From _Vaclav Kotesovec_, Oct 21 2012: (Start)
%F E.g.f.: (265 + 264*x + 135*x^2 + 40*x^3 + 15*x^4 + x^6)/(1-x)^7.
%F D-finite Recurrence: a(n) = (n+7)*a(n-1) - (n-1)*a(n-2), n>=1.
%F a(n) ~ n!*n^6.
%F (End)
%t CoefficientList[Series[-(265 + 264x + 135x^2 + 40x^3 + 15x^4 + x^6)/(x - 1)^7, {x, 0, 20}], x] Range[0, 20]! (* _Vaclav Kotesovec_, Oct 21 2012 *)
%t Differences[Range[0, 23]!, 6] (* _Alonso del Arte_, Nov 10 2018 *)
%o (PARI) x='x+O('x^66); Vec(serlaplace( -(265 +264*x +135*x^2 +40*x^3 +15*x^4 +x^6) / (x-1)^7 )) \\ _Joerg Arndt_, May 04 2013
%o (GAP) a:=[265,2119];; for n in [3..20] do a[n]:=(n+6)*a[n-1]-(n-2)*a[n-2]; od; a; # _Muniru A Asiru_, Nov 23 2018
%o (Magma) I:=[2119, 18806]; [265] cat [n le 2 select I[n] else (n+7)*Self(n-1) - (n-1)*Self(n-2): n in [1..30]]; // _G. C. Greubel_, Nov 23 2018
%o (Sage)
%o f= (265 + 264*x + 135*x^2 + 40*x^3 + 15*x^4 + x^6)/(1-x)^7
%o g=f.taylor(x,0,30)
%o L=g.coefficients()
%o coeffs={c[1]:c[0]*factorial(c[1]) for c in L}
%o coeffs # _G. C. Greubel_, Nov 23 2018
%K nonn,easy
%O 0,1
%A _David W. Wilson_
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