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 A000332 Binomial coefficient binomial(n,4) = n*(n-1)*(n-2)*(n-3)/24. (Formerly M3853 N1578) 341

%I M3853 N1578

%S 0,0,0,0,1,5,15,35,70,126,210,330,495,715,1001,1365,1820,2380,3060,

%T 3876,4845,5985,7315,8855,10626,12650,14950,17550,20475,23751,27405,

%U 31465,35960,40920,46376,52360,58905,66045,73815,82251,91390,101270,111930,123410

%N Binomial coefficient binomial(n,4) = n*(n-1)*(n-2)*(n-3)/24.

%C Number of intersection points of diagonals of convex n-gon where no more than two diagonals intersect at any point in the interior.

%C Also the number of equilateral triangles with vertices in an equilateral triangular array of points with n rows (offset 1), with any orientation. - _Ignacio Larrosa Cañestro_, Apr 09 2002. [See Les Reid link for proof. - _N. J. A. Sloane_, Apr 02 2016]

%C Start from cubane and attach amino acids according to the reaction scheme that describes the reaction between the active sites. See the hyperlink on chemistry. - _Robert G. Wilson v_, Aug 02 2002

%C For n>0 a(n)=(-1/8)*coefficient of x in Zagier's polynomial P_(2n,n). (Zagier's polynomials are used by pari-gp for acceleration of alternating or positive series.)

%C Figurate numbers based on the 4-dimensional regular convex polytope called the regular 4-simplex, pentachoron, 5-cell, pentatope or 4-hypertetrahedron with Schlaefli symbol {3,3,3}. a(n)=((n*(n-1)*(n-2)*(n-3))/4!). - Michael J. Welch (mjw1(AT)ntlworld.com), Apr 01 2004, _R. J. Mathar_, Jul 07 2009

%C Maximal number of crossings that can be created by connecting n vertices with straight lines. - Cameron Redsell-Montgomerie (credsell(AT)uoguelph.ca), Jan 30 2007

%C If X is an n-set and Y a fixed (n-1)-subset of X then a(n) is equal to the number of 4-subsets of X intersecting Y. - _Milan Janjic_, Aug 15 2007

%C Product of four consecutive numbers divided by 24. - _Artur Jasinski_, Dec 02 2007

%C Only prime in this sequence is 5. - _Artur Jasinski_, Dec 02 2007

%C For strings consisting entirely of 0's and 1's, the number of distinct arrangements of four 1's such that 1's are not adjacent. The shortest possible string is 7 characters, of which there is only one solution: 1010101, corresponding to a(5). An eight-character string has 5 solutions, nine has 15, ten has 35 and so on, congruent to A000332. - _Gil Broussard_, Mar 19 2008

%C For a(n)>0, a(n) is pentagonal if and only if 3 does not divide n. All terms belong to the generalized pentagonal sequence (A001318). Cf. A000326, A145919, A145920. - _Matthew Vandermast_, Oct 28 2008

%C Nonzero terms = row sums of triangle A158824. - _Gary W. Adamson_, Mar 28 2009

%C Except for the 4 initial 0's, is equivalent to the partial sums of the tetrahedral numbers A000292. - Jeremy Cahill (jcahill(AT)inbox.com), Apr 15 2009

%C If the first 3 zeros are disregarded, that is, if one looks at binomial(n+3, 4) with n>=0, then it becomes a 'Matryoshka doll' sequence with alpha=0: seq(add(add(add(i,i=alpha..k),k=alpha..n),n=alpha..m),m=alpha..50). - _Peter Luschny_, Jul 14 2009

%C For n>=1, a(n) is the number of n-digit numbers the binary expansion of which contains two runs of 0's. - _Vladimir Shevelev_, Jul 30 2010

%C For n>0, a(n) is the number of crossing set partitions of {1,2,..,n} into n-2 blocks. - _Peter Luschny_, Apr 29 2011

%C The Kn3, Ca3 and Gi3 triangle sums of A139600 are related to the sequence given above, e.g., Gi3(n) = 2*A000332(n+3) - A000332(n+2) + 7*A000332(n+1). For the definitions of these triangle sums, see A180662. - _Johannes W. Meijer_, Apr 29 2011

%C For n > 3, a(n) is the hyper-Wiener index of the path graph on n-2 vertices. - _Emeric Deutsch_, Feb 15 2012

%C Except for the four initial zeros, number of all possible tetrahedra of any size, having the same orientation as the original regular tetrahedron, formed when intersecting the latter by planes parallel to its sides and dividing its edges into n equal parts. - _V.J. Pohjola_, Aug 31 2012

%C a(n+3) is the number of different ways to color the faces (or the vertices) of a regular tetrahedron with n colors if we count mirror images as the same.

%C a(n) = fallfac(n,4)/4! is also the number of independent components of an antisymmetric tensor of rank 4 and dimension n >= 1. Here fallfac is the falling factorial. - _Wolfdieter Lang_, Dec 10 2015

%C Does not satisfy Benford's law [Ross, 2012] - _N. J. A. Sloane_, Feb 12 2017

%C Number of chiral pairs of colorings of the vertices (or faces) of a regular tetrahedron with n available colors. Chiral colorings come in pairs, each the reflection of the other. - _Robert A. Russell_, Jan 22 2020

%C From _Mircea Dan Rus_, Aug 26 2020; (Start)

%C a(n+3) is the number of lattice rectangles (squares included) in a staircase of order n; this is obtained by stacking n rows of consecutive unit lattice squares, aligned either to the left or to the right, which consist of 1, 2, 3, ..., n squares and which are stacked either in the increasing or in the decreasing order of their lengths. Below, there is a staircase or order 4 which contains a(7) = 35 rectangles. [See the Teofil Bogdan and Mircea Dan Rus link, problem 3, under A004320]

%C _

%C |_|_

%C |_|_|_

%C |_|_|_|_

%C |_|_|_|_|

%C (End)

%C Ignoring initial three 0s, it is the number of strings of length n on an ordered alphabet of 5 letters where the characters in the word are in nondecreasing order. E.g., number of length-2 words is 15: aa,ab,ac,ad,ae,bb,bc,bd,be,cc,cd,ce,dd,de,ee. - _Jim Nastos_, Jan 18 2021

%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.

%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 196.

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 74, Problem 8.

%D L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 7.

%D J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D Charles W. Trigg: Mathematical Quickies. New York: Dover Publications, Inc., 1985, p. 53, #191

%H Franklin T. Adams-Watters, <a href="/A000332/b000332.txt">Table of n, a(n) for n = 0..1002</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H Brandy Amanda Barnette, <a href="http://digitalcommons.wku.edu/theses/1484">Counting Convex Sets on Products of Totally Ordered Sets</a>, Masters Theses & Specialist Projects, Paper 1484, 2015.

%H Peter J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

%H Paul Erdős, Norbert Kaufman, R. H. Koch and Arthur Rosenthal, <a href="http://www.jstor.org/stable/2305216">E750 (Interior diagonal points)</a>, Amer. Math. Monthly, 54 (Jun, 1947), p. 344.

%H Th. Grüner, A. Kerber, R. Laue and M. Meringer, <a href="ftp://ftp.mathe2.uni-bayreuth.de/meringer/pdf/MathCombChemSCCE.pdf">Mathematics for Combinatorial Chemistry</a>.

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=254">Encyclopedia of Combinatorial Structures 254</a>.

%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>.

%H Hyun Kwang Kim, <a href="http://dx.doi.org/10.1090/S0002-9939-02-06710-2">On Regular Polytope Numbers</a>, Proc. Amer. Math. Soc., 131 (2002), 65-75.

%H Tim McDevitt and Kathryn Sutcliffe, <a href="https://dx.doi.org/10.5951%2Fmathteacher.110.6.0470">A New Look at an Old Triangle Counting Problem</a>. The Mathematics Teacher. Vol. 110, No. 6 (February 2017), pp. 470-474.

%H Alexsandar Petojevic, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL5/Petojevic/petojevic5.html">The Function vM_m(s; a; z) and Some Well-Known Sequences</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%H Les Reid, <a href="http://people.missouristate.edu/lesreid/POW03_01.html">Counting Triangles in an Array</a>.

%H Les Reid, <a href="/A000332/a000332.pdf">Counting Triangles in an Array</a>. [Cached copy]

%H Luis Manuel Rivera, <a href="http://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv preprint arXiv:1406.3081 [math.CO], 2014.

%H Kenneth A. Ross, <a href="http://www.jstor.org/stable/10.4169/math.mag.85.1.036">First Digits of Squares and Cubes</a>, Math. Mag. 85 (2012) 36-42. doi:10.4169/math.mag.85.1.36.

%H Kirill S. Shardakov and Vladimir P. Bubnov, <a href="http://ceur-ws.org/Vol-2556/paper5.pdf">Stochastic Model of a High-Loaded Monitoring System of Data Transmission Network</a>, Selected Papers of the Models and Methods of Information Systems Research Workshop, CEUR Workshop Proceedings, (St. Petersburg, Russia, 2019), 29-34.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Composition.html">Composition</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PentatopeNumber.html">Pentatope Number</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Pentatope.html">Pentatope</a>.

%H A. F. Y. Zhao, <a href="http://www.emis.de/journals/JIS/VOL17/Zhao/zhao3.html">Pattern Popularity in Multiply Restricted Permutations</a>, Journal of Integer Sequences, 17 (2014), #14.10.3.

%H <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%H <a href="/index/Be#Benford">Index entries for sequences related to Benford's law</a>

%F a(n) = n*(n-1)*(n-2)*(n-3)/24.

%F G.f.: x^4/(1-x)^5. - _Simon Plouffe_ in his 1992 dissertation

%F a(n) = n*a(n-1)/(n-4). - _Benoit Cloitre_, Apr 26 2003, _R. J. Mathar_, Jul 07 2009

%F a(n) = Sum_{k=1..n-3} Sum_{i=1..k} i*(i+1)/2. - _Benoit Cloitre_, Jun 15 2003

%F Convolution of natural numbers {1, 2, 3, 4, ...} and A000217, the triangular numbers {1, 3, 6, 10, ...}. - _Jon Perry_, Jun 25 2003

%F a(n) = A110555(n+1,4). - _Reinhard Zumkeller_, Jul 27 2005

%F a(n+1) = [(n^5-(n-1)^5)-(n^3-(n-1)^3)]/24 - (n^5-(n-1)^5-1)/30; a(n) = A006322(n-2)-A006325(n-1). - Xavier Acloque, Oct 20 2003; _R. J. Mathar_, Jul 07 2009

%F a(4*n+2) = Pyr(n+4, 4*n+2) where the polygonal pyramidal numbers are defined for integers A>2 and B>=0 by Pyr(A, B) = B-th A-gonal pyramid number = [(A-2)*B^3 + 3*B^2 - (A-5)*B]/6; For all positive integers i and the pentagonal number function P(x) = x*(3*x-1)/2: a(3*i-2) = P(P(i)) and a(3*i-1) = P(P(i) + i); 1 + 24*a(n) = (n^2 + 3*n + 1)^2. - _Jonathan Vos Post_, Nov 15 2004

%F First differences of A000389(n). - _Alexander Adamchuk_, Dec 19 2004

%F For n > 3, the sum of the first n-2 tetrahedral numbers (A000292). - Martin Steven McCormick (mathseq(AT)wazer.net), Apr 06 2005 [Corrected by _Doug Bell_, Jun 25 2017]

%F Starting (1, 5, 15, 35, ...), = binomial transform of [1, 4, 6, 4, 1, 0, 0, 0, ...]. - _Gary W. Adamson_, Dec 28 2007

%F Sum_{n>=4} 1/a(n) = 4/3, from the Taylor expansion of (1-x)^3*log(1-x) in the limit x->1. - _R. J. Mathar_, Jan 27 2009

%F A034263(n) = (n+1)*a(n+4) - Sum_{i=0..n+3} a(i). Also A132458(n) = a(n)^2 - a(n-1)^2 for n>0. - _Bruno Berselli_, Dec 29 2010

%F a(0)=0, a(1)=0, a(2)=0, a(3)=0, a(4)=1, a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - _Harvey P. Dale_, Aug 22 2011

%F a(n) = (binomial(n-1,2)^2 - binomial(n-1,2))/6. - _Gary Detlefs_, Nov 20 2011

%F a(n) = Sum_{k=1..n-2} Sum_{i=1..k} i*(n-k-2). - _Wesley Ivan Hurt_, Sep 25 2013

%F a(n) = (A000217(A000217(n - 2) - 1))/3 = ((((n - 2)^2 + (n - 2))/2)^2 - (((n - 2)^2 + (n - 2))/2))/(2*3). - _Raphie Frank_, Jan 16 2014

%F Sum_{n>=0} a(n)/n! = e/24. Sum_{n>=3} a(n)/(n-3)! = 73*e/24. See A067764 regarding the second ratio. - _Richard R. Forberg_, Dec 26 2013

%F Sum_{n>=4} (-1)^(n + 1)/a(n) = 32*log(2) - 64/3 = A242023 = 0.847376444589... . - _Richard R. Forberg_, Aug 11 2014

%F 4/(Sum_{n>=m} 1/a(n)) = A027480(m-3), for m>=4. - _Richard R. Forberg_, Aug 12 2014

%F E.g.f.: x^4*exp(x)/24. - _Robert Israel_, Nov 23 2014

%F a(n+3) = C(n,1) + 3*C(n,2) + 3*C(n,3) + C(n,4). Each term indicates the number of ways to use n colors to color a tetrahedron with exactly 1, 2, 3, or 4 colors.

%F a(n) = A080852(1,n-4). - _R. J. Mathar_, Jul 28 2016

%F G.f.: Starting (1, 5, 14, ...), x/(1-x)^5 = (x * r(x) * r(x^2) * r(x^4) * r(x^8) *...) where r(x) = (1 + x)^5. x/(1-x)^5 = (x * r(x) * r(x^3) * r(x^9) * r(x^27), ...) where r(x) = (1 + x + x^2)^5. x/(1-x)^5 = (x * r(x) * r(x^4) * r(x^16) * r(x^64), * ...) where r(x) = (1 + x + x^2 + x^3)^5; ...(as a conjectured infinite set). - _Gary W. Adamson_, Feb 06 2017

%F From _Robert A. Russell_, Jan 22 2020: (Start)

%F a(n) = A006008(n) - a(n+3) = (A006008(n) - A006003(n)) / 2 = a(n+3) - A006003(n).

%F a(n+3) = A006008(n) - a(n) = (A006008(n) + A006003(n)) / 2 = a(n) + A006003(n).

%F a(n) = A007318(n,4).

%F a(n+3) = A325000(3,n). (End)

%F Product_{n>=5} (1 - 1/a(n)) = cosh(sqrt(15)*Pi/2)/(100*Pi). - _Amiram Eldar_, Jan 21 2021

%e a(5) = 5 from the five independent components of an antisymmetric tensor A of rank 4 and dimension 5, namely A(1,2,3,4), A(1,2,3,5), A(1,2,4,5), A(1,3,4,5) and A(2,3,4,5). See the Dec 10 2015 comment. - _Wolfdieter Lang_, Dec 10 2015

%p A000332 := n->binomial(n,4); [seq(binomial(n,4), n=0..100)];

%t Table[ Binomial[n, 4], {n, 0, 45} ] (* corrected by _Harvey P. Dale_, Aug 22 2011 *)

%t Table[(n-4)(n-3)(n-2)(n-1)/24, {n, 100}] (* _Artur Jasinski_, Dec 02 2007 *)

%t LinearRecurrence[{5,-10,10,-5,1}, {0,0,0,0,1}, 45] (* _Harvey P. Dale_, Aug 22 2011 *)

%t CoefficientList[Series[x^4 / (1 - x)^5, {x, 0, 40}], x] (* _Vincenzo Librandi_, Nov 23 2014 *)

%o (PARI) a(n)=binomial(n,4);

%o (MAGMA) [Binomial(n,4): n in [0..50]]; // _Vincenzo Librandi_, Nov 23 2014

%o (GAP) A000332 := List([1..10^2], n -> Binomial(n, 4)); # _Muniru A Asiru_, Oct 16 2017

%o (Python 3) # Starts at a(3), i.e. computes n*(n+1)*(n+2)*(n+3)/24 which is more in

%o # line with A000217 and A000292.

%o def A000332():

%o x, y, z, u = 1, 1, 1, 1

%o yield 0

%o while True:

%o yield x

%o x, y, z, u = x + y + z + u + 1, y + z + u + 1, z + u + 1, u + 1

%o a = A000332(); print([next(a) for i in range(41)]) # _Peter Luschny_, Aug 03 2019

%Y Cf. A053134, A053126, A000389, A000579-A000582, A075733, A006322, A006325.

%Y Cf. also A000583, A014820, A092181, A092182, A092183.

%Y Cf. A000217, A000292, A007318 (column k = 4).

%Y Cf. A158824.

%Y Cf. A006008 (Number of ways to color the faces (or vertices) of a regular tetrahedron with n colors when mirror images are counted as two).

%Y Cf. A104712 (third column, k=4).

%Y See A269747 for a 3-D analog.

%Y Cf. A006008 (oriented), A006003 (achiral) tetrahedron colorings.

%Y Row 3 of A325000, col. 4 of A007318.

%K nonn,easy,nice

%O 0,6

%A _N. J. A. Sloane_

%E Some formulas that referred to another offset corrected by _R. J. Mathar_, Jul 07 2009

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Last modified April 20 23:46 EDT 2021. Contains 343143 sequences. (Running on oeis4.)