%I #96 Jul 08 2024 10:37:27

%S 0,4,15,36,70,120,189,280,396,540,715,924,1170,1456,1785,2160,2584,

%T 3060,3591,4180,4830,5544,6325,7176,8100,9100,10179,11340,12586,13920,

%U 15345,16864,18480,20196,22015,23940,25974,28120,30381,32760,35260

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

%C Number of independent components of a certain 3-tensor in n-space.

%C 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_{a=1..n} t(a,a,c) = 0 for all c, with a,b,c range 1..n. (3-tensor in n-dimensional space which is symmetric and traceless in one pair of its indices.)

%C Row 2 of the convolution array A213761. - _Clark Kimberling_, Jul 04 2012

%C Also, the number of ways to place two dominoes horizontally in the same row on an (n+2) X (n+2) chessboard. - _Ralf Stephan_, Jun 09 2014

%C Also, the sum of all the numbers in a completely filled n X n tic-tac-toe board with n-1 players using the numbers 0, 1, 2,... n-2. See "Sums of Square Tic Tac Toe Boards that end in a Draw" in links for proof. - _Tanner Robeson_, Aug 23 2020

%C a(n) is the number of degrees of freedom in a tetrahedral cell for a Raviart-Thomas finite element space of order n. - _Matthew Scroggs_, Jan 02 2021

%H Vincenzo Librandi, <a href="/A077414/b077414.txt">Table of n, a(n) for n = 1..1000</a>

%H DefElement, <a href="https://defelement.com/elements/raviart-thomas.html">Raviart-Thomas</a>

%H Sela Fried, <a href="https://arxiv.org/abs/2406.18923">Counting r X s rectangles in nondecreasing and Smirnov words</a>, arXiv:2406.18923 [math.CO], 2024. See p. 9.

%H Tanner Robeson, <a href="/A077414/a077414.pdf">Sums of Square Tic Tac Toe Boards that end in a Draw</a>.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%H <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>.

%F a(n) = n * ( binomial(n+1, 2)-1 ).

%F G.f.: x^2*(4-x)/(1-x)^4.

%F a(n) = n*Sum_{j=2..n} j. - _Zerinvary Lajos_, Sep 12 2006

%F a(1)=0, a(2)=4, a(3)=15, a(4)=36; for n>4, a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - _Harvey P. Dale_, Jun 04 2012

%F a(n) = Sum_{i=1..n-1} i*(3*(n-i)+1). - _Bruno Berselli_, Feb 13 2014

%F a(-n) = -A005564(n). - _Michael Somos_, Jun 09 2014

%F a(n) = A057145(n,n+2). - _R. J. Mathar_, Jul 28 2016

%F a(n) = t(n,t(n,1)) + n, where t(n,k) = n*(n+1)/2 + k*n and t(n,1) = A000096(n). - _Bruno Berselli_, Feb 28 2017

%F a(n) = n^3/2 + n^2/2 - n. - _Tanner Robeson_, Aug 23 2020

%F Sum_{n>=2} 1/a(n) = 7/18. - _Amiram Eldar_, Oct 07 2020

%F Sum_{n>=2} (-1)^n/a(n) = 4*log(2)/3 - 13/18. - _Amiram Eldar_, Feb 22 2022

%F E.g.f.: exp(x)*x^2*(4 + x)/2. - _Stefano Spezia_, Jan 03 2023

%e For n=6, a(6) = 1*(3*5+1)+2*(3*4+1)+3*(3*3+1)+4*(3*2+1)+5*(3*1+1) = 120. - _Bruno Berselli_, Feb 13 2014

%e G.f. = 4*x^2 + 15*x^3 + 36*x^4 + 70*x^5 + 120*x^6 + 189*x^7 + 280*x^8 + ...

%p A077414:=n->n*(n-1)*(n+2)/2: seq(A077414(n), n=1..60); # _Wesley Ivan Hurt_, Apr 09 2017

%t Table[(n (n - 1) (n + 2))/2, {n, 50}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {0, 4, 15, 36}, 50] (* _Harvey P. Dale_, Jun 04 2012 *)

%t CoefficientList[Series[x (4 - x)/(1 - x)^4, {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 14 2014 *)

%o (PARI) a(n)=n*(n-1)*(n+2)/2 \\ _Charles R Greathouse IV_, Oct 07 2015

%o (PARI) concat(0, Vec(x^2*(4-x)/(1-x)^4 + O(x^200))) \\ _Altug Alkan_, Jan 15 2016

%o (Magma) [n*(n-1)*(n+2)/2: n in [1..30]]; // _G. C. Greubel_, Jan 18 2018

%Y Cf. A000096, A005564, A057145, A115067 (first differences), A213761.

%Y Cf. similar sequences of the type m*(m+1)*(m+k)/2 listed in A267370.

%K nonn,easy

%O 1,2

%A _Wolfdieter Lang_, Nov 29 2002