This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A157371 #30 Sep 01 2025 01:45:11
%S A157371 9,8,9,0,-55,-216,-567,-1216,-2295,-3960,-6391,-9792,-14391,-20440,
%T A157371 -28215,-38016,-50167,-65016,-82935,-104320,-129591,-159192,-193591,
%U A157371 -233280,-278775,-330616,-389367,-455616,-529975,-613080,-705591,-808192,-921591,-1046520,-1183735,-1334016,-1498167
%N A157371 a(n) = (n+1)*(9-9*n+5*n^2-n^3).
%C A157371 This is the fourth in a family of sequences that appear in columns on pages 36 and 56 of the reference: (i) sequence n+1, A000029, (ii) sequence (n+1)*(1-n), A147998 and (iii) (n+1)*(5-5*n+2*n^2), A152064.
%C A157371 First differences along columns shown on page 56 of the reference are columns of what is shown on page 36 of the reference. Example: the third column of page 56, A152064, has first differences which constitute the third column p page 36, A140811.
%D A157371 Paul Curtz, Intégration numérique des systèmes différentiels à conditions initiales, Centre de Calcul Scientifique de l'Armement, Note 12, Arcueil (1969).
%H A157371 Vincenzo Librandi, <a href="/A157371/b157371.txt">Table of n, a(n) for n = 0..10000</a>
%H A157371 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A157371 First differences: a(n+1)-a(n) = -A141530(n).
%F A157371 Fourth differences: a(n+4)-4*a(n+3)+6*a(n+2)-4*a(n+1)+a(n) = -24 = -A010863(n).
%F A157371 From _Harvey P. Dale_, Jul 13 2011: (Start)
%F A157371 a(0)=9, a(1)=8, a(2)=9, a(3)=0, a(4)=-55, a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5).
%F A157371 G.f.: (9-37*x+59*x^2-55*x^3)/(1-x)^5. (End)
%F A157371 E.g.f.: (9 - x + x^2 - 2*x^3 - x^4)*exp(x). - _G. C. Greubel_, Feb 02 2018
%t A157371 LinearRecurrence[{5,-10,10,-5,1},{9,8,9,0,-55},40] (* or *) Table[(n+1)(9-9n+5n^2-n^3),{n,0,40}] (* or *) CoefficientList[ Series[ (55x^3- 59x^2+ 37x-9)/ (x-1)^5,{x,0,40}],x] (* _Harvey P. Dale_, Jul 13 2011 *)
%o A157371 (Magma) [(n+1)*(9-9*n+5*n^2-n^3): n in [0..40] ]; // _Vincenzo Librandi_, Jul 14 2011
%o A157371 (PARI) a(n)=(n+1)*(9-9*n+5*n^2-n^3) \\ _Charles R Greathouse IV_, Oct 16 2015
%K A157371 sign,easy,changed
%O A157371 0,1
%A A157371 _Paul Curtz_, Feb 28 2009
%E A157371 Edited, extended by _R. J. Mathar_, Sep 25 2009