A077414 a(n) = n*(n - 1)*(n + 2)/2.

0, 4, 15, 36, 70, 120, 189, 280, 396, 540, 715, 924, 1170, 1456, 1785, 2160, 2584, 3060, 3591, 4180, 4830, 5544, 6325, 7176, 8100, 9100, 10179, 11340, 12586, 13920, 15345, 16864, 18480, 20196, 22015, 23940, 25974, 28120, 30381, 32760, 35260

Offset: 1

Author

Wolfdieter Lang, Nov 29 2002

Keywords

• nonn
• easy

Comments

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

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.)

Row 2 of the convolution array A213761. - Clark Kimberling, Jul 04 2012

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

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

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

Examples

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• DefElement, Raviart-Thomas
• Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 9.
• Tanner Robeson, Sums of Square Tic Tac Toe Boards that end in a Draw.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
• Index to sequences related to polygonal numbers.

Crossrefs

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

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

Programs

Magma

[n*(n-1)*(n+2)/2: n in [1..30]]; // G. C. Greubel, Jan 18 2018

Maple

A077414:=n->n*(n-1)*(n+2)/2: seq(A077414(n), n=1..60); # Wesley Ivan Hurt, Apr 09 2017

Mathematica

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 *)
CoefficientList[Series[x (4 - x)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Feb 14 2014 *)

PARI

a(n)=n*(n-1)*(n+2)/2 \\ Charles R Greathouse IV, Oct 07 2015

PARI

concat(0, Vec(x^2*(4-x)/(1-x)^4 + O(x^200))) \\ Altug Alkan, Jan 15 2016

Formula

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

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

a(n) = n*Sum_{j=2..n} j. - Zerinvary Lajos, Sep 12 2006

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

a(n) = Sum_{i=1..n-1} i*(3*(n-i)+1). - Bruno Berselli, Feb 13 2014

a(-n) = -A005564(n). - Michael Somos, Jun 09 2014

a(n) = A057145(n,n+2). - R. J. Mathar, Jul 28 2016

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

a(n) = n^3/2 + n^2/2 - n. - Tanner Robeson, Aug 23 2020

Sum_{n>=2} 1/a(n) = 7/18. - Amiram Eldar, Oct 07 2020

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

E.g.f.: exp(x)*x^2*(4 + x)/2. - Stefano Spezia, Jan 03 2023