%I #12 Sep 08 2022 08:45:54
%S 1,33,276,1299,4392,11925,27708,57351,108624,191817,320100,509883,
%T 781176,1157949,1668492,2345775,3227808,4358001,5785524,7565667,
%U 9760200,12437733,15674076,19552599,24164592,29609625,35995908
%N Generating function A(5,t)(1+t+t^2)/(1-t)^6, where A(5,t) is an Eulerian polynomial.
%C The Eulerian polynomials A(n,t) are here defined in accordance with the Digital Library of Mathematical Functions, Table 26.14.1.
%C Sums of 3 consecutive fifth powers: a(n) = (n-1)^5+n^5+(n+1)^5. - _Bruno Berselli_, Jun 24 2013
%H OEIS Wiki, <a href="http://oeis.org/wiki/Eulerian_polynomials"> Eulerian polynomials</a>.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F From _Bruno Berselli_, Jun 24 2013: (Start)
%F G.f.: (1 + 27*x + 93*x^2 + 118*x^3 + 93*x^4 + 27*x^5 + x^6) / (1 - x)^6.
%F a(n) = n*(3*n^4 + 20*n^2 + 10) for n>0, a(0)=1. (End)
%F a(0)=1, a(1)=33, a(2)=276, a(3)=1299, a(4)=4392, a(5)=11925, a(6)=27708; for n>6, a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6). - _Harvey P. Dale_, Apr 10 2015
%p gfA179995 := proc(t) local i;
%p add([1,27,93,118,93,27,1][i+1]*t^i,i=0..5)/(1-t)^6 end:
%p seq(coeff(series(gfA179995(t),t,24),t,j),j=0..16);
%t Join[{1}, Table[n (3 n^4 + 20 n^2 + 10), {n, 30}]] (* _Bruno Berselli_, Jun 24 2013 *)
%t LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 33, 276, 1299, 4392, 11925, 27708}, 30] (* _Harvey P. Dale_, Apr 10 2015 *)
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+27*x+93*x^2+118*x^3+93*x^4+27*x^5+x^6)/(1-x)^6)); // _Bruno Berselli_, Jun 24 2013
%Y Cf. A158799, A008486, A005918, A027602, A160827 which have generating functions of type A(n, t)(1+t+t^2)/(1-t)^(n+1).
%K nonn,easy
%O 0,2
%A _Peter Luschny_, Aug 05 2010