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A004320 a(n) = n*(n+1)*(n+2)^2/6. 10

%I

%S 0,3,16,50,120,245,448,756,1200,1815,2640,3718,5096,6825,8960,11560,

%T 14688,18411,22800,27930,33880,40733,48576,57500,67600,78975,91728,

%U 105966,121800,139345,158720,180048,203456,229075,257040,287490,320568,356421,395200

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

%C Consider the set B(n) = {1,2,3,...n}. Let a(0) = 0. Then a(n) = Sum [ b(i)^2 - b(j)^2] for all i, j = 1 to n, b(i) belongs to B(n). E.g., a(3) = (3^2-1^2) + (3^2-2^2) + (2^2-1^2) = 16. - _Amarnath Murthy_, Jun 01 2001

%C Partial sums of A016061. - _J. M. Bergot_, Jun 18 2013

%C For n>=3, a(n-2) is the number of permutations of n symbols that 3-commute with an n-cycle (see A233440 for definition). - _Luis Manuel Rivera Martínez_, Feb 24 2014

%C a(n) is the sum of all pairs with repetitions allowed drawn from the set of triangular numbers from A000217(0) to A000217(n). This is similar to A027480 but uses triangular numbers instead of the integers. Example for n=2: 0+1, 0+3, 1+1, 1+3, 3+3 gives sum of 16=a(2). - _J. M. Bergot_, Mar 23 2016

%H Vincenzo Librandi, <a href="/A004320/b004320.txt">Table of n, a(n) for n = 0..10000</a>

%H Luis Manuel Rivera, <a href="http://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv preprint arXiv:1406.3081 [math.CO], 2014.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F G.f.: x*(3+x)/(1-x)^5. - _Paul Barry_, Feb 27 2003

%F a(n) = (n+2)*A000292(n). - _Zerinvary Lajos_, May 26 2006

%F a(n) = A047929(n+2)/6. - _Zerinvary Lajos_, May 09 2007

%F a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5). - _Wesley Ivan Hurt_, Oct 28 2014

%F Sum_{n>=1} 1/a(n) = Pi^2/2 - 9/2. - _Jaume Oliver Lafont_, Jul 13 2017

%p [seq ((n+2)*(binomial(n+2,3)), n=0..45)]; # _Zerinvary Lajos_, May 26 2006

%t Table[n (n + 1) (n + 2)^2/6, {n, 0, 40}] (* _Wesley Ivan Hurt_, Oct 28 2014 *)

%o (MAGMA) [n*(n+1)*(n+2)^2/6: n in [0..40] ]; // _Vincenzo Librandi_, Aug 19 2011

%o (PARI) a(n)=n*(n+1)*(n+2)^2/6 \\ _Charles R Greathouse IV_, Jun 18 2013

%Y Cf. A016061, A047929, A233440.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_

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Last modified January 25 16:42 EST 2020. Contains 331245 sequences. (Running on oeis4.)