%I #51 Apr 29 2023 07:02:03

%S 1,6,19,46,94,172,290,460,695,1010,1421,1946,2604,3416,4404,5592,7005,

%T 8670,10615,12870,15466,18436,21814,25636,29939,34762,40145,46130,

%U 52760,60080,68136,76976,86649,97206,108699,121182,134710,149340

%N Let r, s, t be three permutations of the set {1,2,3,..,n}; a(n) = value of Sum_{i=1..n} r(i)*s(i)*t(i), with r={1,2,3,..,n}; s={n,n-1,..,1} and t={n,n-2,n-4,...,1,...,n-3,n-1}.

%C See A070735 for the minimal values for these products. This sequence is an upper bound. The third permutation 't'= ceiling(abs(range(n-1/2,-n,-2))) is such that it associates its smallest factor with the largest factor of the product 'r'*'s'.

%C We observe that is the transform of A002717 by the following transform T: T(u_0,u_1,u_2,u_3,...) = (u_0,u_0+u_1, u_0+u_1+u_2, u_0+u_1+u_2+u_3+u_4,...). In other words, v_p = Sum_{k=0..p} u_k and the g.f. phi_v of v is given by phi_v = phi_u/(1-z). - _Richard Choulet_, Jan 28 2010

%H Vincenzo Librandi, <a href="/A070893/b070893.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,0,5,-4,1).

%F G.f.: x*(1+2*x)/((1+x)*(1-x)^5). - _Michael Somos_, Apr 07 2003

%F a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5) + 3. If sequence is also defined for n <= 0 by this equation, then a(n)=0 for -3 <= n <= 0 and a(n)=A082289(-n) for n <= -4. - _Michael Somos_, Apr 07 2003

%F a(n) = (1/96)*(2*n*(n+2)*(3*n^2+10*n+4)+3*(-1)^n-3). a(n) - a(n-2) = A002411(n). - _Bruno Berselli_, Aug 26 2011

%e {1,2,3,4,5,6,7}*{7,6,5,4,3,2,1}*{7,5,3,1,2,4,6} gives {49,60,45,16,30,48,42}, with sum 290, so a(7)=290.

%t Table[Plus@@(Range[n]*Range[n, 1, -1]*Ceiling[Abs[Range[n-1/2, -n, -2]]]), {n, 49}];

%t (* or *)

%t CoefficientList[Series[ -(1+2x)/(-1+x)^5/(1+x), {x, 0, 48}], x]//Flatten

%o (PARI) a(n)=sum(i=1,n,i*(n+1-i)*ceil(abs(n+3/2-2*i)))

%o (PARI) a(n)=polcoeff(if(n<0,x^4*(2+x)/((1+x)*(1-x)^5),x*(1+2*x)/((1+x)*(1-x)^5))+x*O(x^abs(n)),abs(n))

%o (Magma) [(1/96)*(2*n*(n+2)*(3*n^2+10*n+4)+3*(-1)^n-3): n in [1..40]]; // _Vincenzo Librandi_, Aug 26 2011

%Y Cf. A070735, A082289. a(n)=A082290(2n-2).

%Y Cf. A000034, A032766, A006578, A002717. - _Richard Choulet_, Jan 28 2010

%Y Cf. A002717 (first differences). - _Bruno Berselli_, Aug 26 2011

%Y Column k=3 of A166278. - _Alois P. Heinz_, Nov 02 2012

%K nonn,easy

%O 1,2

%A _Wouter Meeussen_, May 22 2002