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A077415 a(n) = n*(n+2)*(n-2)/3. 14
0, 5, 16, 35, 64, 105, 160, 231, 320, 429, 560, 715, 896, 1105, 1344, 1615, 1920, 2261, 2640, 3059, 3520, 4025, 4576, 5175, 5824, 6525, 7280, 8091, 8960, 9889, 10880, 11935, 13056, 14245, 15504, 16835, 18240, 19721, 21280, 22919, 24640, 26445 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

COMMENTS

a(n) is the number of independent components of a 3-tensor t(a,b,c) which satisfies t(a,b,c)=t(b,a,c) and sum(t(a,a,c),a=1..n)=0 for all c and t(a,b,c)+t(b,c,a)+t(c,a,b)=0, with a,b,c range 1..n. (3-tensor in n-dimensional space which is symmetric and traceless in one pair of its indices and satisfies the cyclic identity.)

Number of standard tableaux of shape (n-1,2,1) (n>=3). - Emeric Deutsch, May 13 2004

Zero followed by partial sums of A028387, starting at n=1. - Klaus Brockhaus, Oct 21 2008

For n>=4, a(n-1) is the number of permutations of 1,2...,n with the distribution of up (1) - down (0) elements 0...0101 (the first n-4 zeros), or, the same, a(n-1) is up-down coefficient {n,5} (see comment in A060351). - Vladimir Shevelev, Feb 14 2014

For n>=3, a(n) equals the second immanant of the (n-1) X (n-1) tridiagonal matrix with 2's along the main diagonal, and 1's along the superdiagonal and the subdiagonal. - John M. Campbell, Jan 08 2016

LINKS

Table of n, a(n) for n=2..43.

M. R. Sepanski, On Divisibility of Convolutions of Central Binomial Coefficients, Electronic Journal of Combinatorics, 21 (1) 2014, #P1.32.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

FORMULA

a(n) = n*(n+2)*(n-2)/3 = A077414(n) - binomial(n+2,3) = A077414(n) - A000292(n-1).

G.f.: x^3*(5 - 4*x + x^2)/(1-x)^4.

a(n) = A084990(n-1) - 1. - Reinhard Zumkeller, Aug 20 2007

a(n) = sum((-1)^i*2^(n-2*i-1)*binomial(n-i-1, i)*(n-2*i-2), i=0..[(n-1)/2]). - John M. Campbell, Jan 08 2016

MAPLE

seq((n^3-4*n)/3, n=2..35); # Zerinvary Lajos, Jan 20 2007

MATHEMATICA

lst={}; s=0; Do[s+=n^2-n-1; AppendTo[lst, s], {n, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Sep 30 2008 *)

Table[((n-1)*n*(n+1)-(n-1)-n-(n+1))/3, {n, -6, 60}] (* Vladimir Joseph Stephan Orlovsky, Nov 18 2009 *)

Print[Table[Sum[(-1)^i*2^(n-2*i-1)*Binomial[n-i-1, i]*(n-2*i-2), {i, 0, Floor[(n-1)/2]}], {n, 2, 100}]] ;  (* John M. Campbell, Jan 08 2016 *)

LinearRecurrence[{4, -6, 4, -1}, {0, 5, 16, 35}, 50] (* Vincenzo Librandi, Jan 09 2016 *)

PROG

(PARI) {a=0; print1(a, ", "); for(n=1, 42, print1(a=a+n+(n+1)^2, ", "))} \\ Klaus Brockhaus, Oct 21 2008

(PARI) concat(0, Vec(x^3*(5-4*x+x^2)/(1-x)^4 + O(x^100))) \\ Altug Alkan, Jan 08 2015

(MAGMA) [n*(n+2)*(n-2)/3: n in [2..50]]; /* or */ I:=[0, 5, 16, 35]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jan 09 2016

CROSSREFS

Cf. A028387 (first differences), A033275 (partial sums).

Sequence in context: A246697 A098404 A190970 * A234362 A108966 A184635

Adjacent sequences:  A077412 A077413 A077414 * A077416 A077417 A077418

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Nov 29 2002

STATUS

approved

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Last modified June 27 15:10 EDT 2017. Contains 288790 sequences.