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A023043 6th differences of factorial numbers.

%I A023043 #33 Sep 08 2022 08:44:47
%S A023043 265,2119,18806,183822,1965624,22852200,287250480,3884393520,
%T A023043 56255149440,869007242880,14266826784000,248112809683200,
%U A023043 4557208289356800,88166812070937600,1792259345728051200,38195370237024000000,851609625265631232000,19827505082582765568000
%N A023043 6th differences of factorial numbers.
%H A023043 Vincenzo Librandi, <a href="/A023043/b023043.txt">Table of n, a(n) for n = 0..200</a>
%H A023043 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a>
%H A023043 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F A023043 From _Vaclav Kotesovec_, Oct 21 2012: (Start)
%F A023043 E.g.f.: (265 + 264*x + 135*x^2 + 40*x^3 + 15*x^4 + x^6)/(1-x)^7.
%F A023043 D-finite Recurrence: a(n) = (n+7)*a(n-1) - (n-1)*a(n-2), n>=1.
%F A023043 a(n) ~ n!*n^6.
%F A023043 (End)
%t A023043 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 A023043 Differences[Range[0, 23]!, 6] (* _Alonso del Arte_, Nov 10 2018 *)
%o A023043 (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 A023043 (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 A023043 (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 A023043 (Sage)
%o A023043 f= (265 + 264*x + 135*x^2 + 40*x^3 + 15*x^4 + x^6)/(1-x)^7
%o A023043 g=f.taylor(x,0,30)
%o A023043 L=g.coefficients()
%o A023043 coeffs={c[1]:c[0]*factorial(c[1]) for c in L}
%o A023043 coeffs # _G. C. Greubel_, Nov 23 2018
%K A023043 nonn,easy
%O A023043 0,1
%A A023043 _David W. Wilson_