%I #202 Jul 08 2024 10:38:36
%S 0,0,0,3,15,45,105,210,378,630,990,1485,2145,3003,4095,5460,7140,9180,
%T 11628,14535,17955,21945,26565,31878,37950,44850,52650,61425,71253,
%U 82215,94395,107880,122760,139128,157080,176715,198135,221445,246753,274170,303810,335790
%N Tritriangular numbers: a(n) = binomial(binomial(n,2),2) = n*(n+1)*(n-1)*(n-2)/8.
%C "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).
%C Several different versions of this sequence are possible, beginning with either one, two or three 0's.
%C 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
%C 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
%C Partial sums of A027480. - _J. M. Bergot_, Jul 09 2013
%C 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
%C 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
%C 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
%C Also the number of 4-cycles in the complete graph K_{n+1}. - _Eric W. Weisstein_, Mar 13 2018
%C 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
%D Arthur T. Benjamin and Jennifer Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 154.
%D Louis Comtet, Advanced Combinatorics, Reidel, 1974, Problem 1, page 72.
%D Richard P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.5, case k=2.
%H William A. Tedeschi, <a href="/A050534/b050534.txt">Table of n, a(n) for n = 0..10000</a>
%H Serhat Bulut and Oktay Erkan Temizkan, <a href="https://web.archive.org/web/20160708101054/http://matematikproje.com/dosyalar/7e1cdSubset_smallest_elements_Sum.pdf">Subset Sum Problem</a>, Jan 20 2015.
%H Alexander Burstein, Sergey Kitaev and Toufik Mansour, <a href="https://web.archive.org/web/20210415000955/http://puma.dimai.unifi.it/19_2_3/3.pdf">Partially ordered patterns and their combinatorial interpretations</a>, PU. M. A., Vol. 19, No. 2-3 (2008), pp. 27-38.
%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 Frank Harary and Bennet Manvel, <a href="http://dml.cz/dmlcz/126802">On the number of cycles in a graph</a>, Matemat. casop. 21 (1971) 55-63, Theorem 1 for 4-cycles in complete graph.
%H Louis H. Kauffman, <a href="http://arxiv.org/abs/1109.1085">Non-Commutative Worlds-Classical Constraints, Relativity and the Bianchi Identity</a>, arXiv preprint arXiv:1109.1085 [math-ph], 2011. (See Appendix)
%H Alexander Kreinin, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Kreinin/kreinin4.html">Integer Sequences and Laplace Continued Fraction</a>, Journal of Integer Sequences, Vol. 19 (2016), Article 16.6.2.
%H Ronald Orozco López, <a href="https://www.researchgate.net/publication/350397609_Solution_of_the_Differential_Equation_ykeay_Special_Values_of_Bell_Polynomials_and_ka-Autonomous_Coefficients">Solution of the Differential Equation y^(k)= e^(a*y), Special Values of Bell Polynomials and (k,a)-Autonomous Coefficients</a>, Universidad de los Andes (Colombia 2021).
%H Frank Ruskey and Jennifer Woodcock, <a href="http://dx.doi.org/10.1007/978-3-642-25011-8_23">The Rand and block distances of pairs of set partitions</a>, in Combinatorial Algorithms, 287-299, Lecture Notes in Comput. Sci., 7056, Springer, Heidelberg, 2011.
%H C. N. Schmall, <a href="http://www.jstor.org/stable/2973734">Problem 432</a>, The American Mathematical Monthly, Vol. 22, No. 4 (1915), p. 130.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CompleteGraph.html">Complete Graph</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TritriangularNumber.html">Tritriangular Number</a>.
%H Chai Wah Wu, <a href="http://arxiv.org/abs/1407.5663">Graphs whose normalized Laplacian matrices are separable as density matrices in quantum mechanics</a>, arXiv:1407.5663 [quant-ph], 2014.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F a(n) = 3*binomial(n+1, 4) = 3*A000332(n+1).
%F From _Vladeta Jovovic_, May 03 2002: (Start)
%F Recurrence: a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
%F G.f.: 3*x^3 / (1-x)^5. (End)
%F a(n+1) = T(T(n)) - T(n); a(n+2) = T(T(n)+n) where T is A000217. - _Jon Perry_, Jun 11 2003
%F a(n+1) = T(n)^2 - T(T(n)) where T is A000217. - _Jon Perry_, Jul 23 2003
%F a(n) = T(T(n-1)-1) where T is A000217. - _Jon E. Schoenfield_, Dec 14 2014
%F a(n) = 3*C(n, 4) + 3*C(n, 3), for n>3.
%F From _Alexander Adamchuk_, Apr 11 2006: (Start)
%F a(n) = (1/2)*Sum_{k=1..n} k*(k-1)*(k-2).
%F a(n) = A033487(n-2)/2, n>1.
%F a(n) = C(n-1,2)*C(n+1,2)/2, n>2. (End)
%F a(n) = A052762(n+1)/8. - _Zerinvary Lajos_, Apr 26 2007
%F 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
%F E.g.f.: x^3*exp(x)*(4+x)/8. - _Robert Israel_, Nov 01 2015
%F a(n) = Sum_{k=1..n} Sum_{i=1..k} (n-i-1)*(n-k). - _Wesley Ivan Hurt_, Sep 12 2017
%F a(n) = A001296(n-1) - A000292(n-1). - _Cyril Damamme_, Feb 26 2018
%F Sum_{n>=3} 1/a(n) = 4/9. - _Vaclav Kotesovec_, May 01 2018
%F a(n) = A006528(n) - A002817(n) = (A006528(n) - A002411(n)) / 2 = A002817(n) - A002411(n). - _Robert A. Russell_, Oct 20 2020
%F Sum_{n>=3} (-1)^(n+1)/a(n) = 32*log(2)/3 - 64/9. - _Amiram Eldar_, Jan 09 2022
%F a(n) = Sum_{k=1..2} (-1)^(k+1)*binomial(n,2-k)*binomial(n,2+k). - _Gerry Martens_, Oct 09 2022
%e For a(3)=3, the chiral pairs of square colorings are AABC-AACB, ABBC-ACBB, and ABCC-ACCB. - _Robert A. Russell_, Oct 20 2020
%p [seq(binomial(n+1,4)*3,n=0..40)]; # _Zerinvary Lajos_, Jul 18 2006
%t Table[Binomial[Binomial[n, 2], 2], {n, 0, 50}] (* _Stefan Steinerberger_, Apr 08 2006 *)
%t LinearRecurrence[{5, -10, 10, -5, 1}, {0, 0, 0, 3, 15}, 40] (* _Harvey P. Dale_, Dec 14 2011 *)
%t (* Start from _Eric W. Weisstein_, Mar 13 2018 *)
%t Binomial[Binomial[Range[0, 20], 2], 2]
%t Nest[Binomial[#, 2] &, Range[0, 20], 2]
%t Nest[PolygonalNumber[# - 1] &, Range[0, 20], 2]
%t CoefficientList[Series[3 x^3/(1 - x)^5, {x, 0, 20}], x]
%t (* End *)
%o (Sage) [(binomial(binomial(n,2),2)) for n in range(0, 39)] # _Zerinvary Lajos_, Nov 30 2009
%o (PARI) a(n)=n*(n+1)*(n-1)*(n-2)/8 \\ _Charles R Greathouse IV_, Nov 20 2012
%o (Magma) [3*Binomial(n+1, 4): n in [0..40]]; // _Vincenzo Librandi_, Feb 14 2015
%o (PARI) x='x+O('x^100); concat([0, 0, 0], Vec(3*x^3/(1-x)^5)) \\ _Altug Alkan_, Nov 01 2015
%o (GAP) List([0..40],n->3*Binomial(n+1,4)); # _Muniru A Asiru_, Mar 20 2018
%Y Cf. A000217, A000332, A033487, A107394, A034827, A210569, Second column of triangle A001498.
%Y Cf. similar sequences listed in A241765.
%Y Cf. (square colorings) A006528 (oriented), A002817 (unoriented), A002411 (achiral),
%Y Row 2 of A325006 (orthoplex facets, orthotope vertices) and A337409 (orthotope edges, orthoplex ridges).
%Y Row 4 of A293496 (cycles of n colors using k or fewer colors).
%K easy,nice,nonn
%O 0,4
%A Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 29 1999
%E Additional comments from Antreas P. Hatzipolakis, May 03 2002