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A050534
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Tritriangular numbers: a(n)=binomial(binomial(n,2),2) = n(n + 1)(n - 1)(n - 2)/8.
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28
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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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)". - The American Mathematical Monthly 22(1915) 130 by C. N. Schmall
Several different versions of this sequence are possible, beginning with either one, two or three 0's.
a(n)=A052762(n+1)/8. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 26 2007
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 R. Janjic (agnus(AT)blic.net), Dec 28 2007
Number of unique ways to select 2 pairs of objects from a set of N objects, when order doesn't matter. For example, with n = 4, the 3 possibilities are (12)(34), (13)(24), and (14)(23). - Brian Parsonnet, Jan 03 2012
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REFERENCES
| A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 154.
L. Comtet, Advanced Combinatorics, Reidel, 1974, Problem 1, page 72.
L. H. Kauffman, Non-Commutative Worlds-Classical Constraints, Relativity and the Bianchi Identity, Arxiv preprint arXiv:1109.1085, 2011. (See Appendix)
Frank Ruskey and Jennifer Woodcock, The Rand and block distances of pairs of set partitions, in International Workshop on Combinatorial Algorithms, Victoria, 2011. LNCS, to appear.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.5, case k=2.
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LINKS
| William A. Tedeschi, Table of n, a(n) for n=0..10000
Index entries for sequences related to linear recurrences with constant coefficients
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FORMULA
| a(n) = 3*binomial(n+3, 4) = 3*A000332(n+3). [This produces 0, 3, 15, 45, ...]
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/(1-x)^5. - Vladeta Jovovic (vladeta(AT)eunet.rs), May 03 2002
Define T(n)=n*(n+1)/2, then a(n)=T(T(n))-T(n) and also a(n+1)=T(T(n)+n) - Jon Perry (perry(AT)globalnet.co.uk), Jun 11 2003
Also a(n)=T(n)^2-T(T(n)) - Jon Perry (perry(AT)globalnet.co.uk), Jul 23 2003
a(n) = 3C(n, 4) + 3C(n, 3), for n>3.
a(n) = Sum[(k*(k-1)*(k-2)),{k,1,n}]/2. a(n) = A033487(n-2)/2, n>1. a(n) = C(n-1,2)*C(n+1,2)/2, n>2. - Alexander Adamchuk (alex(AT)kolmogorov.com), Apr 11 2006
a(n)=numbperm (n,4)/8, n>=1 . - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 26 2007
a(n) = (4x^4 - 4x^3 - x^2 + x)/2 where x = floor(n/2)*(-1)^n for n >= 0. [From William A. Tedeschi (fynmun(AT)att.net), Aug 24 2010]
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MAPLE
| [seq(binomial(n, 4)*3, n=1..40)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 18 2006
seq(numbperm (n, 4)/8, n=1..39); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 26 2007
seq(sum(sum(sum(m, k=0..l), l=0..m), m=1..n), n=-2..36); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 26 2008
a:=n->sum((n-j)^3-n+j, j=1..n): seq(a(n)/2, n=0..38); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 26 2008
a:=n->add(binomial(n, 2)+add(binomial(n, 2), j=0..n), j=0..n):seq(a(n)/4, n=-1..30); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Aug 26 2008]
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MATHEMATICA
| Table[Binomial[Binomial[n, 2], 2], {n, 0, 50}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 08 2006
Table[Sum[(k*(k-1)*(k-2)), {k, 1, n}]/2, {n, 0, 60}] - Alexander Adamchuk (alex(AT)kolmogorov.com), Apr 11 2006
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 0, 0, 3, 15}, 40] (* From Harvey P. Dale, Dec 14 2011 *)
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PROG
| (Other) sage: [(binomial(binomial(n, 2), 2)) for n in xrange(0, 39)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 30 2009]
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CROSSREFS
| Cf. A000217, A000332.
Second column of triangle A001498.
Cf. A033487, A107394, A033487, A034827.
Sequence in context: A161400 A112810 A094191 * A048099 A030505 A074355
Adjacent sequences: A050531 A050532 A050533 * A050535 A050536 A050537
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KEYWORD
| easy,nice,nonn
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AUTHOR
| Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 29 1999
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EXTENSIONS
| Additional comments from Antreas P. Hatzipolakis, May 03, 2002
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