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A080957 Expansion of (5 - 9*x + 6*x^2)/(1-x)^4.

%I A080957 #38 May 09 2025 05:19:22
%S A080957 5,11,20,34,55,85,126,180,249,335,440,566,715,889,1090,1320,1581,1875,
%T A080957 2204,2570,2975,3421,3910,4444,5025,5655,6336,7070,7859,8705,9610,
%U A080957 10576,11605,12699,13860,15090,16391,17765,19214,20740,22345,24031,25800
%N A080957 Expansion of (5 - 9*x + 6*x^2)/(1-x)^4.
%C A080957 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 A080957 Danny Rorabaugh, <a href="/A080957/b080957.txt">Table of n, a(n) for n = 0..10000</a>
%H A080957 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A080957 a(n) = 3!*(C(n+1, 1) - C(n+2, 2)/2 + C(n+3, 3)/3) = (2*n^3 + 3*n^2 + 31*n + 30)/6.
%F A080957 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 A080957 a(n+1) = a(n) + A117951(n+1), a(0) = 5. - _Altug Alkan_, Sep 28 2015
%F A080957 E.g.f.: (1/6)*(30 + 36*x + 9*x^2 + 2*x^3)*exp(x). - _G. C. Greubel_, May 08 2025
%t A080957 CoefficientList[Series[(5-9 x +6 x^2)/(1-x)^4, {x, 0, 45}], x] (* _Vincenzo Librandi_ Sep 07 2015 *)
%t A080957 LinearRecurrence[{4,-6,4,-1},{5,11,20,34},50] (* _Harvey P. Dale_, Dec 23 2018 *)
%o A080957 (PARI) Vec((5-9*x+6*x^2)/(1-x)^4 + O(x^60)) \\ _Michel Marcus_, Sep 06 2015
%o A080957 (Magma) [(2*n^3+3*n^2+31*n+30)/6: n in [0..50]]; // _Vincenzo Librandi_, Sep 07 2015
%o A080957 (PARI) a(n)=(2*n^3 + 3*n^2 + 31*n + 30)/6;
%o A080957 vector(40, n, a(n-1)) \\ _Altug Alkan_, Sep 28 2015
%o A080957 (SageMath)
%o A080957 def A080957(n): return (2*n^3 +3*n^2 +31*n +30)//6
%o A080957 print([A080957(n) for n in range(51)]) # _G. C. Greubel_, May 08 2025
%Y A080957 Cf. A080956, A117951.
%K A080957 easy,nonn
%O A080957 0,1
%A A080957 _Paul Barry_, Mar 01 2003