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A127878 a(n) = n^4 + 4*n^3 + 12*n^2 + 24*n + 24.

%I A127878 #25 Sep 08 2022 08:45:29
%S A127878 24,65,168,393,824,1569,2760,4553,7128,10689,15464,21705,29688,39713,
%T A127878 52104,67209,85400,107073,132648,162569,197304,237345,283208,335433,
%U A127878 394584,461249,536040,619593,712568,815649,929544,1054985,1192728
%N A127878 a(n) = n^4 + 4*n^3 + 12*n^2 + 24*n + 24.
%C A127878 Generating polynomial is Schur's polynomial of 4-degree. Schur's polynomials n degree are n-th first term of series expansion of e^x function. All polynomials are non-reducible and belonging to the An alternating Galois transitive group if n is divisible by 4 or to Sn symmetric Galois Group in other case (proof Schur, 1930).
%H A127878 G. C. Greubel, <a href="/A127878/b127878.txt">Table of n, a(n) for n = 0..10000</a>
%H A127878 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A127878 Integral representation in terms of incomplete Gamma function : a(n)= Exp[n]Gamma[5,n], where Gamma[5,n]= Integrate[x^4 Exp[ -x], {x, n, +infinity}]. - _N-E. Fahssi_, Jan 25 2008
%F A127878 G.f.: (24 -55*x +83*x^2 -37*x^3 +9*x^4)/(1-x)^5. - _Colin Barker_, Apr 02 2012
%F A127878 E.g.f.: (24 + 41*x + 31*x^2 + 10*x^3 + x^4)*exp(x). - _G. C. Greubel_, Apr 29 2018
%p A127878 seq(n^4+4*n^3+12*n^2+24*n+24,n=0..40); # _Muniru A Asiru_, Apr 30 2018
%t A127878 Table[24 + 24*n + 12*n^2 + 4*n^3 + n^4, {n, 0, 50}]
%t A127878 LinearRecurrence[{5, -10, 10, -5, 1}, {24, 65, 168, 393, 824}, 50] (* _G. C. Greubel_, Apr 29 2018 *)
%o A127878 (PARI) for(n=0,30, print1(n^4 +4*n^3 +12*n^2 +24*n +24, ", ")) \\ _G. C. Greubel_, Apr 29 2018
%o A127878 (Magma) [n^4 +4*n^3 +12*n^2 +24*n +24: n in [0..30]]; // _G. C. Greubel_, Apr 29 2018
%o A127878 (GAP) List([0..40],n->n^4+4*n^3+12*n^2+24*n+24); # _Muniru A Asiru_, Apr 30 2018
%Y A127878 Cf. A127873-A127883.
%K A127878 nonn,easy
%O A127878 0,1
%A A127878 _Artur Jasinski_, Feb 04 2007