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A037237 Expansion of (3 + x^2) / (1 - x)^4.

%I A037237 #43 Sep 08 2022 08:44:52
%S A037237 3,12,31,64,115,188,287,416,579,780,1023,1312,1651,2044,2495,3008,
%T A037237 3587,4236,4959,5760,6643,7612,8671,9824,11075,12428,13887,15456,
%U A037237 17139,18940,20863,22912,25091,27404
%N A037237 Expansion of (3 + x^2) / (1 - x)^4.
%C A037237 This sequence is the partial sums of A058331. - _J. M. Bergot_, May 31 2012
%H A037237 Vincenzo Librandi, <a href="/A037237/b037237.txt">Table of n, a(n) for n = 0..1000</a>
%H A037237 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1)
%F A037237 a(n) = Sum_{k=0..n} (2*(k+1)^2 + 1). - _Mike Warburton_, Jul 07 2007, Sep 07 2007
%F A037237 a(n) = (n+1)*(2*n^2 + 7*n + 9)/3. - _R. J. Mathar_, Mar 29 2010
%F A037237 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Vincenzo Librandi_, Jun 21 2012
%F A037237 E.g.f.: (1/3)*(9 + 27*x + 15*x^2 + 2*x^3)*exp(x). - _G. C. Greubel_, Jul 22 2017
%t A037237 CoefficientList[Series[(3+x^2)/(1-x)^4,{x,0,50}],x]  (* _Harvey P. Dale_, Mar 06 2011 *)
%t A037237 LinearRecurrence[{4,-6,4,-1},{3,12,31,64},40] (* _Vincenzo Librandi_ Jun 21 2012 *)
%o A037237 (Magma) I:=[3, 12, 31, 64]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)- Self(n-4): n in [1..40]]; // _Vincenzo Librandi_, Jun 21 2012
%o A037237 (PARI) x='x+O('x^50); Vec((3+x^2)/(1-x)^4) \\ _G. C. Greubel_, Jul 22 2017
%K A037237 nonn,easy
%O A037237 0,1
%A A037237 _N. J. A. Sloane_