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%I #32 Sep 08 2022 08:45:11
%S 1,13,67,227,602,1358,2730,5034,8679,14179,22165,33397,48776,69356,
%T 96356,131172,175389,230793,299383,383383,485254,607706,753710,926510,
%U 1129635,1366911,1642473,1960777,2326612,2745112
%N a(n) = Sum_{i=1..n} i^2*t(i), where t = A000217.
%C This sequence is related to A001296 by a(n) = n*A001296(n) - Sum_{i=0..n-1} A001296(i) with n>0. - _Bruno Berselli_, Jan 21 2013
%H Vincenzo Librandi, <a href="/A086689/b086689.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F a(n) = n*(n+1)*(n+2)*(12*n^2+9*n-1)/120.
%F G.f.: x*(1+7*x+4*x^2) / (x-1)^6. - _R. J. Mathar_, Sep 15 2012
%F a(n) = 6*a(n-1)-15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6). - _Wesley Ivan Hurt_, Nov 19 2014
%F a(n) = Sum_{i=1..n} ( i*Sum_{k=1..i} i*k ). - _Wesley Ivan Hurt_, Nov 19 2014
%e a(4) = 227 = 1^2*A000217(1)+2^2*A000217(2)+3^2*A000217(3)+4^2*A000217(4).
%p A086689:=n->n*(n+1)*(n+2)*(12*n^2+9*n-1)/120: seq(A086689(n), n=1..40); # _Wesley Ivan Hurt_, Nov 19 2014
%t Table[n (n + 1) (n + 2) (12 n^2 + 9 n - 1)/120, {n, 40}] (* _Wesley Ivan Hurt_, Nov 19 2014 *)
%t CoefficientList[Series[(1 + 7 x + 4 x^2) / (x - 1)^6, {x, 0, 50}], x] (° _Vincenzo Librandi_, Nov 20 2014 °)
%o (PARI) t(n)=n*(n+1)/2 for(i=1,30,print1(","sum(j=1,i,j^2*t(i))))
%o (Magma) [n*(n+1)*(n+2)*(12*n^2+9*n-1)/120 : n in [1..40]]; // _Wesley Ivan Hurt_, Nov 19 2014
%Y Cf. A001296.
%K nonn,easy
%O 1,2
%A _Jon Perry_, Jul 28 2003
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