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A070893 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}. 10
1, 6, 19, 46, 94, 172, 290, 460, 695, 1010, 1421, 1946, 2604, 3416, 4404, 5592, 7005, 8670, 10615, 12870, 15466, 18436, 21814, 25636, 29939, 34762, 40145, 46130, 52760, 60080, 68136, 76976, 86649, 97206, 108699, 121182, 134710, 149340 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

See A070735 for the minimal values for these products. This series 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'.

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 another terms v_p=sum(u_k,k=0..p) and the G.f phi_v of v is given by: phi_v=phi_u/(1-z). - Richard Choulet, Jan 28 2010

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Index entries for two-way infinite sequences

Index entries for linear recurrences with constant coefficients, signature (4,-5,0,5,-4,1).

FORMULA

G.f.: x*(1+2*x)/((1+x)*(1-x)^5). - Michael Somos, Apr 07 2003

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

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

EXAMPLE

{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.

MAPLE

with(combinat):a[0]:=0:for n from 1 to 50 do a[n]:=stirling2(n+2, n)-a[n-1] od: seq(a[n], n=1..38); # Zerinvary Lajos, Mar 17 2008

MATHEMATICA

Table[Plus@@(Range[n]*Range[n, 1, -1]*Ceiling[Abs[Range[n-1/2, -n, -2]]]), {n, 49}]; or CoefficientList[Series[ -(1+2x)/(-1+x)^5/(1+x), {x, 0, 48}], x]//Flatten

PROG

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

(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))

(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

CROSSREFS

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

Cf. A000034, A032766, A006578, A002717. - Richard Choulet, Jan 28 2010

Cf. A002717 (first differences). - Bruno Berselli, Aug 26 2011

Column k=3 of A166278. - Alois P. Heinz, Nov 02 2012

Sequence in context: A272707 A266938 A005712 * A272047 A267829 A027963

Adjacent sequences:  A070890 A070891 A070892 * A070894 A070895 A070896

KEYWORD

nonn,easy

AUTHOR

Wouter Meeussen, May 22 2002

STATUS

approved

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Last modified December 7 23:05 EST 2016. Contains 278900 sequences.