A050534 Tritriangular numbers: a(n) = binomial(binomial(n,2),2) = n*(n+1)*(n-1)*(n-2)/8.

0, 0, 0, 3, 15, 45, 105, 210, 378, 630, 990, 1485, 2145, 3003, 4095, 5460, 7140, 9180, 11628, 14535, 17955, 21945, 26565, 31878, 37950, 44850, 52650, 61425, 71253, 82215, 94395, 107880, 122760, 139128, 157080, 176715, 198135, 221445, 246753, 274170, 303810, 335790

Offset: 0

Author

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 29 1999

Keywords

• easy
• nice
• nonn

Comments

"There are n straight lines in a plane, no two of which are parallel and no three of which are concurrent. Their points of intersection being joined, show that the number of new lines drawn is (1/8)n(n-1)(n-2)(n-3)." (Schmall, 1915).

Several different versions of this sequence are possible, beginning with either one, two or three 0's.

If Y is a 3-subset of an n-set X then, for n>=6, a(n-4) is the number of (n-6)-subsets of X which have exactly one element in common with Y. - Milan Janjic, Dec 28 2007

Number of distinct ways to select 2 pairs of objects from a set of n+1 objects, when order doesn't matter. For example, with n = 3 (4 objects), the 3 possibilities are (12)(34), (13)(24), and (14)(23). - Brian Parsonnet, Jan 03 2012

Partial sums of A027480. - J. M. Bergot, Jul 09 2013

For the set {1,2,...,n}, the sum of the 2 smallest elements of all subsets with 3 elements is a(n) (see Bulut et al. link). - Serhat Bulut, Jan 20 2015

a(n) is also the number of subgroups of S_{n+1} (the symmetric group on n+1 elements) that are isomorphic to D_4 (the dihedral group of order 8). - Geoffrey Critzer, Sep 13 2015

a(n) is the coefficient of x1^(n-3)*x2^2 in exponential Bell polynomial B_{n+1}(x1,x2,...) (number of ways to select 2 pairs among n+1 objects, see above), hence its link with A000292 and A001296 (see formula). - Cyril Damamme, Feb 26 2018

Also the number of 4-cycles in the complete graph K_{n+1}. - Eric W. Weisstein, Mar 13 2018

Number of chiral pairs of colorings of the 4 edges or vertices of a square using n or fewer colors. Each member of a chiral pair is a reflection, but not a rotation, of the other. - Robert A. Russell, Oct 20 2020

Examples

			For a(3)=3, the chiral pairs of square colorings are AABC-AACB, ABBC-ACBB, and ABCC-ACCB. - _Robert A. Russell_, Oct 20 2020
		

References

• Arthur T. Benjamin and Jennifer Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 154.
• Louis Comtet, Advanced Combinatorics, Reidel, 1974, Problem 1, page 72.
• Richard P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.5, case k=2.

Links

• William A. Tedeschi, Table of n, a(n) for n = 0..10000
• Serhat Bulut and Oktay Erkan Temizkan, Subset Sum Problem, Jan 20 2015.
• Alexander Burstein, Sergey Kitaev and Toufik Mansour, Partially ordered patterns and their combinatorial interpretations, PU. M. A., Vol. 19, No. 2-3 (2008), pp. 27-38.
• Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 9.
• Frank Harary and Bennet Manvel, On the number of cycles in a graph, Matemat. casop. 21 (1971) 55-63, Theorem 1 for 4-cycles in complete graph.
• Louis H. Kauffman, Non-Commutative Worlds-Classical Constraints, Relativity and the Bianchi Identity, arXiv preprint arXiv:1109.1085 [math-ph], 2011. (See Appendix)
• Alexander Kreinin, Integer Sequences and Laplace Continued Fraction, Journal of Integer Sequences, Vol. 19 (2016), Article 16.6.2.
• Ronald Orozco López, Solution of the Differential Equation y^(k)= e^(a*y), Special Values of Bell Polynomials and (k,a)-Autonomous Coefficients, Universidad de los Andes (Colombia 2021).
• Frank Ruskey and Jennifer Woodcock, The Rand and block distances of pairs of set partitions, in Combinatorial Algorithms, 287-299, Lecture Notes in Comput. Sci., 7056, Springer, Heidelberg, 2011.
• C. N. Schmall, Problem 432, The American Mathematical Monthly, Vol. 22, No. 4 (1915), p. 130.
• Eric Weisstein's World of Mathematics, Complete Graph.
• Eric Weisstein's World of Mathematics, Graph Cycle.
• Eric Weisstein's World of Mathematics, Tritriangular Number.
• Chai Wah Wu, Graphs whose normalized Laplacian matrices are separable as density matrices in quantum mechanics, arXiv:1407.5663 [quant-ph], 2014.
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A000217, A000332, A033487, A107394, A034827, A210569, Second column of triangle A001498.

Cf. similar sequences listed in A241765.

Cf. (square colorings) A006528 (oriented), A002817 (unoriented), A002411 (achiral),

Row 2 of A325006 (orthoplex facets, orthotope vertices) and A337409 (orthotope edges, orthoplex ridges).

Row 4 of A293496 (cycles of n colors using k or fewer colors).

Programs

GAP

List([0..40],n->3*Binomial(n+1,4)); # Muniru A Asiru, Mar 20 2018

Magma

[3*Binomial(n+1, 4): n in [0..40]]; // Vincenzo Librandi, Feb 14 2015

Maple

[seq(binomial(n+1,4)*3,n=0..40)]; # Zerinvary Lajos, Jul 18 2006

Mathematica

Table[Binomial[Binomial[n, 2], 2], {n, 0, 50}] (* Stefan Steinerberger, Apr 08 2006 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 0, 0, 3, 15}, 40] (* Harvey P. Dale, Dec 14 2011 *)
(* Start from Eric W. Weisstein, Mar 13 2018 *)
Binomial[Binomial[Range[0, 20], 2], 2]
Nest[Binomial[#, 2] &, Range[0, 20], 2]
Nest[PolygonalNumber[# - 1] &, Range[0, 20], 2]
CoefficientList[Series[3 x^3/(1 - x)^5, {x, 0, 20}], x]
(* End *)

PARI

a(n)=n*(n+1)*(n-1)*(n-2)/8 \\ Charles R Greathouse IV, Nov 20 2012

PARI

x='x+O('x^100); concat([0, 0, 0], Vec(3*x^3/(1-x)^5)) \\ Altug Alkan, Nov 01 2015

Sage

[(binomial(binomial(n,2),2)) for n in range(0, 39)] # Zerinvary Lajos, Nov 30 2009

Formula

a(n) = 3*binomial(n+1, 4) = 3*A000332(n+1).

From Vladeta Jovovic, May 03 2002: (Start)

Recurrence: a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).

G.f.: 3*x^3 / (1-x)^5. (End)

a(n+1) = T(T(n)) - T(n); a(n+2) = T(T(n)+n) where T is A000217. - Jon Perry, Jun 11 2003

a(n+1) = T(n)^2 - T(T(n)) where T is A000217. - Jon Perry, Jul 23 2003

a(n) = T(T(n-1)-1) where T is A000217. - Jon E. Schoenfield, Dec 14 2014

a(n) = 3*C(n, 4) + 3*C(n, 3), for n>3.

From Alexander Adamchuk, Apr 11 2006: (Start)

a(n) = (1/2)*Sum_{k=1..n} k*(k-1)*(k-2).

a(n) = A033487(n-2)/2, n>1.

a(n) = C(n-1,2)*C(n+1,2)/2, n>2. (End)

a(n) = A052762(n+1)/8. - Zerinvary Lajos, Apr 26 2007

a(n) = (4x^4 - 4x^3 - x^2 + x)/2 where x = floor(n/2)*(-1)^n for n >= 0. - William A. Tedeschi, Aug 24 2010

E.g.f.: x^3*exp(x)*(4+x)/8. - Robert Israel, Nov 01 2015

a(n) = Sum_{k=1..n} Sum_{i=1..k} (n-i-1)*(n-k). - Wesley Ivan Hurt, Sep 12 2017

a(n) = A001296(n-1) - A000292(n-1). - Cyril Damamme, Feb 26 2018

Sum_{n>=3} 1/a(n) = 4/9. - Vaclav Kotesovec, May 01 2018

a(n) = A006528(n) - A002817(n) = (A006528(n) - A002411(n)) / 2 = A002817(n) - A002411(n). - Robert A. Russell, Oct 20 2020

Sum_{n>=3} (-1)^(n+1)/a(n) = 32*log(2)/3 - 64/9. - Amiram Eldar, Jan 09 2022

a(n) = Sum_{k=1..2} (-1)^(k+1)*binomial(n,2-k)*binomial(n,2+k). - Gerry Martens, Oct 09 2022

Extensions

Additional comments from Antreas P. Hatzipolakis, May 03 2002