A172080 - OEIS

%I #33 Sep 08 2022 08:45:50

%S 0,1,37,190,590,1415,2891,5292,8940,14205,21505,31306,44122,60515,

%T 81095,106520,137496,174777,219165,271510,332710,403711,485507,579140,

%U 685700,806325,942201,1094562,1264690,1453915,1663615,1895216,2150192

%N a(n) = n*(12*n^3 + 10*n^2 - 9*n - 7)/6.

%C The sequence is related to A172078 by a(n) = n*A172078(n) - Sum_{i=0..n-1} A172078(i).

%C This is the case d=8 in the identity n^2*(n+1)*(2*d*n-2*d+3)/6 - Sum_{k=0..n-1} k*(k+1)*(2*d*k - 2*d + 3)/6 = n*(n+1)*(3*d*n^2 - d*n + 4*n - 2*d + 2)/12. - _Bruno Berselli_, May 07 2010, Feb 26 2011

%H Vincenzo Librandi, <a href="/A172080/b172080.txt">Table of n, a(n) for n = 0..1000</a>

%H B. Berselli, A description of the recursive method in Comments lines: website <a href="http://www.lanostra-matematica.org/2008/12/sequenze-numeriche-e-procedimenti.html">Matem@ticamente</a> (in Italian).

%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) = n*(n+1)*(12*n^2 - 2*n - 7)/6.

%F G.f.: x*(1 + 32*x + 15*x^2)/(1-x)^5. - _Bruno Berselli_, Feb 26 2011

%F E.g.f.: x*(6 + 105*x + 82*x^2 + 12*x^3)*exp(x)/6. - _G. C. Greubel_, Aug 30 2019

%p seq(n*(n+1)*(12*n^2 -2*n -7)/6, n=0..40); # _G. C. Greubel_, Aug 30 2019

%t CoefficientList[Series[x(1+32x+15x^2)/(1-x)^5, {x, 0, 40}], x] (* _Vincenzo Librandi_, Jan 01 2014 *)

%t Table[n*(n+1)*(12*n^2 -2*n -7)/6, {n,0,40}] (* _G. C. Greubel_, Aug 30 2019 *)

%o (Magma) [(12*n^4+10*n^3-9*n^2-7*n)/6: n in [0..50]]; // _Vincenzo Librandi_, Jan 01 2014

%o (PARI) vector(40, n, n*(n-1)*(12*(n-1)^2 -2*n -5)/6) \\ _G. C. Greubel_, Aug 30 2019

%o (Sage) [n*(n+1)*(12*n^2 -2*n -7)/6 for n in (0..40)] # _G. C. Greubel_, Aug 30 2019

%o (GAP) List([0..40], n-> n*(n+1)*(12*n^2 -2*n -7)/6); _G. C. Greubel_, Aug 30 2019

%Y Cf. A172078.

%K nonn,easy

%O 0,3

%A _Vincenzo Librandi_, Jan 25 2010