0,4

"There are n straight lines in a plane, no two of which are parallel and no three of which are concurrent. Their points of interesection 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

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.

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.5, case k=2.

Index entries for sequences related to linear recurrences with constant coefficients

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) = A107394(n-3)/2 = 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

[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]

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

(Other) sage: [(binomial(binomial(n, 2), 2)) for n in xrange(0, 39)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 30 2009]

Cf. A000217, A000332.

Second column of triangle A001498.

Cf. A033487, A107394.

Cf. A033487, A034827.

Sequence in context: A161400 A112810 A094191 this_sequence A048099 A030505 A074355

Adjacent sequences: A050531 A050532 A050533 this_sequence A050535 A050536 A050537

easy,nice,nonn,new

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

Additional comments from Antreas P. Hatzipolakis, May 03, 2002