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a(n) = n*(n+1)*(n^2+n+1)/2.
6

%I #76 Sep 08 2022 08:45:20

%S 0,3,21,78,210,465,903,1596,2628,4095,6105,8778,12246,16653,22155,

%T 28920,37128,46971,58653,72390,88410,106953,128271,152628,180300,

%U 211575,246753,286146,330078,378885,432915,492528,558096,630003,708645,794430

%N a(n) = n*(n+1)*(n^2+n+1)/2.

%C This sequence is related to A085461 by 3*A085461(n) = n*a(n) - Sum_{i=0..n-1} a(i) for n>0. - _Bruno Berselli_, Dec 27 2010

%C Subsequence of the triangular numbers A000217, see formulas below. - _David James Sycamore_, Jul 31 2018

%H G. C. Greubel, <a href="/A110450/b110450.txt">Table of n, a(n) for n = 0..5000</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) = Sum_{k=0..n} A110449(n,k), sums of rows in triangle A110449.

%F From _Bruno Berselli_, Dec 27 2010: (Start)

%F G.f.: 3*x*(1 + x)^2/(1 - x)^5.

%F a(n) = A014105(A000217(n)). (End)

%F a(n) = Sum_{i=1..n*(n+1)} i. - _Wesley Ivan Hurt_, Sep 27 2013

%F a(n) = Sum_{i=0..n} i*(2*i^2+1), and these are the partial sums of A061317. - _Bruno Berselli_, Feb 09 2017

%F a(n) = t(n,t(n,A000217(n))), where t(n,k) = n*(n+1)/2 + k*n and k=0. - _Bruno Berselli_, Feb 28 2017

%F E.g.f.: (x/2)*(6 + 15*x + 8*x^2 + x^3)*exp(x). - _G. C. Greubel_, Aug 24 2017

%F a(n) = A000217(n*(n+1)). - _David James Sycamore_, Jul 31 2018

%F a(n) = A000217(2*A000217(n)) = A000217(A002378(n)). - _Alois P. Heinz_, Jul 31 2018

%F a(n) = A002378(n)+A062392(n). - _R. J. Mathar_, Mar 23 2021

%F a(n) = 3*A006325(n+1) .- _R. J. Mathar_, Mar 23 2021

%p A110450:=n->n*(n+1)*(n^2+n+1)/2; seq(A110450(k), k=0..50); # _Wesley Ivan Hurt_, Sep 27 2013

%t Table[n (n + 1) (n^2 + n + 1)/2, {n, 0, 100}] (* _Wesley Ivan Hurt_, Sep 27 2013 *)

%t CoefficientList[Series[-3 x (x^2 + 2 x + 1)/(x - 1)^5, {x, 0, 36}], x] (* or *)

%t LinearRecurrence[{5, -10, 10, -5, 1}, {0, 3, 21, 78, 210}, 36] (* _Robert G. Wilson v_, Jul 31 2018 *)

%o (Magma)[n*(n+1)*(n^2+n+1)/2: n in [0..40]]; // _Vincenzo Librandi_, Dec 26 2010

%o (PARI) a(n)=n*(n+1)*(n^2+n+1)/2 \\ _Charles R Greathouse IV_, Oct 16 2015

%o (GAP) List([0..40],n->n*(n+1)*(n^2+n+1)/2); # _Muniru A Asiru_, Aug 02 2018

%Y Cf. A000217, A002378, A061317, A085461, A110449, A014105.

%K nonn,easy

%O 0,2

%A _Reinhard Zumkeller_, Jul 21 2005