%I #32 Sep 08 2022 08:45:09
%S 5,11,20,34,55,85,126,180,249,335,440,566,715,889,1090,1320,1581,1875,
%T 2204,2570,2975,3421,3910,4444,5025,5655,6336,7070,7859,8705,9610,
%U 10576,11605,12699,13860,15090,16391,17765,19214,20740,22345,24031,25800
%N Expansion of (5 - 9*x + 6*x^2)/(1-x)^4.
%C Coefficient of x in the polynomial 6*(C(n,0) + C(n+1,1)x + C(n+2,2)x(x-1)/2 + C(n+3,3)x(x-1)(x-2)/6).
%H Danny Rorabaugh, <a href="/A080957/b080957.txt">Table of n, a(n) for n = 0..10000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = 3!(C(n+1, 1)-C(n+2, 2)/2+C(n+3, 3)/3) = (2n^3 + 3n^2 + 31n + 30)/6.
%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n>3. - _Vincenzo Librandi_, Sep 07 2015
%F a(n+1) = a(n) + A117951(n+1), a(0) = 5. - _Altug Alkan_, Sep 28 2015
%t CoefficientList[Series[(5 - 9 x + 6 x^2)/(1 - x)^4, {x, 0, 45}], x] (* _Vincenzo Librandi_ Sep 07 2015 *)
%t LinearRecurrence[{4,-6,4,-1},{5,11,20,34},50] (* _Harvey P. Dale_, Dec 23 2018 *)
%o (PARI) Vec((5-9*x+6*x^2)/(1-x)^4 + O(x^60)) \\ _Michel Marcus_, Sep 06 2015
%o (Magma) [(2*n^3+3*n^2+31*n+30)/6: n in [0..50]]; // _Vincenzo Librandi_, Sep 07 2015
%o (PARI) a(n)=(2*n^3 + 3*n^2 + 31*n + 30)/6;
%o vector(40, n, a(n-1)) \\ _Altug Alkan_, Sep 28 2015
%Y Cf. A080956.
%K easy,nonn
%O 0,1
%A _Paul Barry_, Mar 01 2003