A157411 - OEIS

%I #15 Sep 08 2022 08:45:42

%S -19,11,-19,251,1901,6731,17261,36731,69101,119051,191981,294011,

%T 431981,613451,846701,1140731,1505261,1950731,2488301,3129851,3887981,

%U 4776011,5807981,6998651,8363501,9918731,11681261,13668731,15899501,18392651

%N a(n) = 30*n^4 - 120*n^3 + 120*n^2 - 19.

%C These are the numerators in column j=4 of the array in A140825 (reference p. 36).

%C The other columns in A140825 are represented by A000012, A005408, A140811 and A141530.

%C The link between these columns is given by the first differences: a(n+1) - a(n) = 30*A141530(n), where 30 = A027760(4) = A027760(3) = A027642(4) = A002445(2), then for j=3, A141530(n+1) - A141530(n) = A140070(2)*A140811(n).

%D P. Curtz, Integration numerique des systemes differentiels a conditions initiales, Centre de Calcul Scientifique de l'Armement, Note 12, Arcueil (1969).

%H Vincenzo Librandi, <a href="/A157411/b157411.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5, -10, 10, -5, 1).

%F a(n)= 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).

%F G.f.: (-19 + 106*x - 264*x^2 + 646*x^3 + 251*x^4)/(1-x)^5.

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + 720. Fourth differences are constant, 720.

%t Table[30n^4-120n^3+120n^2-19,{n,0,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{-19,11,-19,251,1901},40] (* _Harvey P. Dale_, Mar 08 2015 *)

%o (Magma) [30*n^4 - 120*n^3 + 120*n^2 - 19: n in [0..50]]; // _Vincenzo Librandi_, Aug 07 2011

%o (PARI) a(n)=30*n^4-120*n^3+120*n^2-19 \\ _Charles R Greathouse IV_, Oct 16 2015

%K sign,easy

%O 0,1

%A _Paul Curtz_, Feb 28 2009

%E Edited, one index corrected and extended by _R. J. Mathar_, Sep 17 2009