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%I A000332 M3853 N1578 #458 Mar 08 2025 15:25:54
%S A000332 0,0,0,0,1,5,15,35,70,126,210,330,495,715,1001,1365,1820,2380,3060,
%T A000332 3876,4845,5985,7315,8855,10626,12650,14950,17550,20475,23751,27405,
%U A000332 31465,35960,40920,46376,52360,58905,66045,73815,82251,91390,101270,111930,123410
%N A000332 Binomial coefficient binomial(n,4) = n*(n-1)*(n-2)*(n-3)/24.
%C A000332 Number of intersection points of diagonals of convex n-gon where no more than two diagonals intersect at any point in the interior.
%C A000332 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 A000332 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 A000332 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 A000332 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 A000332 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 A000332 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 A000332 Product of four consecutive numbers divided by 24. - _Artur Jasinski_, Dec 02 2007
%C A000332 The only prime in this sequence is 5. - _Artur Jasinski_, Dec 02 2007
%C A000332 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 A000332 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 A000332 Nonzero terms = row sums of triangle A158824. - _Gary W. Adamson_, Mar 28 2009
%C A000332 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 A000332 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 A000332 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 A000332 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 A000332 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 A000332 For n > 3, a(n) is the hyper-Wiener index of the path graph on n-2 vertices. - _Emeric Deutsch_, Feb 15 2012
%C A000332 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 A000332 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 A000332 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 A000332 Does not satisfy Benford's law [Ross, 2012] - _N. J. A. Sloane_, Feb 12 2017
%C A000332 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 A000332 From _Mircea Dan Rus_, Aug 26 2020: (Start)
%C A000332 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 A000332 _
%C A000332 |_|_
%C A000332 |_|_|_
%C A000332 |_|_|_|_
%C A000332 |_|_|_|_|
%C A000332 (End)
%C A000332 a(n+4) 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
%C A000332 From _Tom Copeland_, Jun 07 2021: (Start)
%C A000332 Aside from the zeros, this is the fifth diagonal of the Pascal matrix A007318, the only nonvanishing diagonal (fifth) of the matrix representation IM = (A132440)^4/4! of the differential operator D^4/4!, when acting on the row vector of coefficients of an o.g.f., or power series.
%C A000332 M = e^{IM} is the matrix of coefficients of the Appell sequence p_n(x) = e^{D^4/4!} x^n = e^{b. D} x^n = (b. + x)^n = Sum_{k=0..n} binomial(n,k) b_n x^{n-k}, where the (b.)^n = b_n have the e.g.f. e^{b.t} = e^{t^4/4!}, which is that for A025036 aerated with triple zeros, the first column of M.
%C A000332 See A099174 and A000292 for analogous relationships for the third and fourth diagonals of the Pascal matrix. (End)
%C A000332 For integer m and positive integer r >= 3, the polynomial a(n) + a(n + m) + a(n + 2*m) + ... + a(n + r*m) in n has its zeros on the vertical line Re(n) = (3 - r*m)/2 in the complex plane. - _Peter Bala_, Jun 02 2024
%D A000332 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 A000332 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 196.
%D A000332 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 74, Problem 8.
%D A000332 John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 70.
%D A000332 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 A000332 Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §8.6 Figurate Numbers, p. 294.
%D A000332 J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
%D A000332 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000332 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A000332 Charles W. Trigg, Mathematical Quickies, New York: Dover Publications, Inc., 1985, p. 53, #191.
%D A000332 David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 127.
%H A000332 Franklin T. Adams-Watters, Table of n, a(n) for n = 0..1002
%H A000332 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A000332 Brandy Amanda Barnette, Counting Convex Sets on Products of Totally Ordered Sets, Masters Theses & Specialist Projects, Paper 1484, 2015.
%H A000332 Gaston A. Brouwer, Jonathan Joe, Abby A. Noble, and Matt Noble, Problems on the Triangular Lattice, arXiv:2405.12321 [math.CO], 2024. Mentions this sequence.
%H A000332 Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000332 Ömür Deveci and Anthony G. Shannon, Some aspects of Neyman triangles and Delannoy arrays, Mathematica Montisnigri (2021) Vol. L, 36-43.
%H A000332 Paul Erdős, Norbert Kaufman, R. H. Koch, and Arthur Rosenthal, E750 (Interior diagonal points), Amer. Math. Monthly, 54 (Jun, 1947), p. 344.
%H A000332 Th. Grüner, A. Kerber, R. Laue, and M. Meringer, Mathematics for Combinatorial Chemistry.
%H A000332 Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See p. 4.
%H A000332 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 254.
%H A000332 Milan Janjic, Two Enumerative Functions.
%H A000332 Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2002), 65-75.
%H A000332 Iva Kodrnja and Helena Koncul, Polynomials vanishing on a basis of S_m(Gamma_0(N)), Glasnik Matematički (2024) Vol. 59, No. 79, 313-325. See p. 324.
%H A000332 Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See pp. 13, 15.
%H A000332 Tim McDevitt and Kathryn Sutcliffe, A New Look at an Old Triangle Counting Problem. The Mathematics Teacher. Vol. 110, No. 6 (February 2017), pp. 470-474.
%H A000332 Rajesh Kumar Mohapatra and Tzung-Pei Hong, On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences, Mathematics (2022) Vol. 10, No. 7, 1161.
%H A000332 Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.
%H A000332 Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A000332 Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
%H A000332 Les Reid, Counting Triangles in an Array.
%H A000332 Les Reid, Counting Triangles in an Array. [Cached copy]
%H A000332 Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
%H A000332 Kenneth A. Ross, First Digits of Squares and Cubes, Math. Mag. 85 (2012) 36-42.
%H A000332 Kirill S. Shardakov and Vladimir P. Bubnov, Stochastic Model of a High-Loaded Monitoring System of Data Transmission Network, Selected Papers of the Models and Methods of Information Systems Research Workshop, CEUR Workshop Proceedings, (St. Petersburg, Russia, 2019), 29-34.
%H A000332 Eric Weisstein's World of Mathematics, Composition.
%H A000332 Eric Weisstein's World of Mathematics, Pentatope Number.
%H A000332 Eric Weisstein's World of Mathematics, Pentatope.
%H A000332 A. F. Y. Zhao, Pattern Popularity in Multiply Restricted Permutations, Journal of Integer Sequences, 17 (2014), #14.10.3.
%H A000332 Index to sequences related to pyramidal numbers.
%H A000332 Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
%H A000332 Index entries for sequences related to Benford's law.
%F A000332 a(n) = n*(n-1)*(n-2)*(n-3)/24.
%F A000332 G.f.: x^4/(1-x)^5. - _Simon Plouffe_ in his 1992 dissertation
%F A000332 a(n) = n*a(n-1)/(n-4). - _Benoit Cloitre_, Apr 26 2003, _R. J. Mathar_, Jul 07 2009
%F A000332 a(n) = Sum_{k=1..n-3} Sum_{i=1..k} i*(i+1)/2. - _Benoit Cloitre_, Jun 15 2003
%F A000332 Convolution of natural numbers {1, 2, 3, 4, ...} and A000217, the triangular numbers {1, 3, 6, 10, ...}. - _Jon Perry_, Jun 25 2003
%F A000332 a(n) = A110555(n+1,4). - _Reinhard Zumkeller_, Jul 27 2005
%F A000332 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 A000332 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 A000332 First differences of A000389(n). - _Alexander Adamchuk_, Dec 19 2004
%F A000332 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 A000332 Starting (1, 5, 15, 35, ...), = binomial transform of [1, 4, 6, 4, 1, 0, 0, 0, ...]. - _Gary W. Adamson_, Dec 28 2007
%F A000332 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 A000332 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 A000332 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5); a(0)=0, a(1)=0, a(2)=0, a(3)=0, a(4)=1. - _Harvey P. Dale_, Aug 22 2011
%F A000332 a(n) = (binomial(n-1,2)^2 - binomial(n-1,2))/6. - _Gary Detlefs_, Nov 20 2011
%F A000332 a(n) = Sum_{k=1..n-2} Sum_{i=1..k} i*(n-k-2). - _Wesley Ivan Hurt_, Sep 25 2013
%F A000332 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 A000332 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 A000332 Sum_{n>=4} (-1)^(n+1)/a(n) = 32*log(2) - 64/3 = A242023 = 0.847376444589... . - _Richard R. Forberg_, Aug 11 2014
%F A000332 4/(Sum_{n>=m} 1/a(n)) = A027480(m-3), for m>=4. - _Richard R. Forberg_, Aug 12 2014
%F A000332 E.g.f.: x^4*exp(x)/24. - _Robert Israel_, Nov 23 2014
%F A000332 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 A000332 a(n) = A080852(1,n-4). - _R. J. Mathar_, Jul 28 2016
%F A000332 From _Gary W. Adamson_, Feb 06 2017: (Start)
%F A000332 G.f.: Starting (1, 5, 14, ...), x/(1-x)^5 can be written
%F A000332 as (x * r(x) * r(x^2) * r(x^4) * r(x^8) * ...) where r(x) = (1+x)^5;
%F A000332 as (x * r(x) * r(x^3) * r(x^9) * r(x^27) * ...) where r(x) = (1+x+x^2)^5;
%F A000332 as (x * r(x) * r(x^4) * r(x^16) * r(x^64) * ...) where r(x) = (1+x+x^2+x^3)^5;
%F A000332 ... (as a conjectured infinite set). (End)
%F A000332 From _Robert A. Russell_, Jan 22 2020: (Start)
%F A000332 a(n) = A006008(n) - a(n+3) = (A006008(n) - A006003(n)) / 2 = a(n+3) - A006003(n).
%F A000332 a(n+3) = A006008(n) - a(n) = (A006008(n) + A006003(n)) / 2 = a(n) + A006003(n).
%F A000332 a(n) = A007318(n,4).
%F A000332 a(n+3) = A325000(3,n). (End)
%F A000332 Product_{n>=5} (1 - 1/a(n)) = cosh(sqrt(15)*Pi/2)/(100*Pi). - _Amiram Eldar_, Jan 21 2021
%e A000332 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 A000332 := n->binomial(n,4); [seq(binomial(n,4), n=0..100)];
%t A000332 Table[ Binomial[n, 4], {n, 0, 45} ] (* corrected by _Harvey P. Dale_, Aug 22 2011 *)
%t A000332 Table[(n-4)(n-3)(n-2)(n-1)/24, {n, 100}] (* _Artur Jasinski_, Dec 02 2007 *)
%t A000332 LinearRecurrence[{5,-10,10,-5,1}, {0,0,0,0,1}, 45] (* _Harvey P. Dale_, Aug 22 2011 *)
%t A000332 CoefficientList[Series[x^4 / (1 - x)^5, {x, 0, 40}], x] (* _Vincenzo Librandi_, Nov 23 2014 *)
%o A000332 (PARI) a(n)=binomial(n,4);
%o A000332 (Magma) [Binomial(n,4): n in [0..50]]; // _Vincenzo Librandi_, Nov 23 2014
%o A000332 (GAP) A000332 := List([1..10^2], n -> Binomial(n, 4)); # _Muniru A Asiru_, Oct 16 2017
%o A000332 (Python)
%o A000332 # Starts at a(3), i.e. computes n*(n+1)*(n+2)*(n+3)/24
%o A000332 # which is more in line with A000217 and A000292.
%o A000332 def A000332():
%o A000332 x, y, z, u = 1, 1, 1, 1
%o A000332 yield 0
%o A000332 while True:
%o A000332 yield x
%o A000332 x, y, z, u = x + y + z + u + 1, y + z + u + 1, z + u + 1, u + 1
%o A000332 a = A000332(); print([next(a) for i in range(41)]) # _Peter Luschny_, Aug 03 2019
%o A000332 (Python)
%o A000332 print([n*(n-1)*(n-2)*(n-3)//24 for n in range(50)])
%o A000332 # _Gennady Eremin_, Feb 06 2022
%Y A000332 binomial(n, k): A161680 (k = 2), A000389 (k = 5), A000579 (k = 6), A000580 (k = 7), A000581 (k = 8), A000582 (k = 9).
%Y A000332 Cf. A053134, A053126, A075733, A006322, A006325.
%Y A000332 Cf. also A000583, A014820, A092181, A092182, A092183.
%Y A000332 Cf. A000217, A000292, A007318 (column k = 4).
%Y A000332 Cf. A158824.
%Y A000332 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 A000332 Cf. A104712 (third column, k=4).
%Y A000332 See A269747 for a 3-D analog.
%Y A000332 Cf. A006008 (oriented), A006003 (achiral) tetrahedron colorings.
%Y A000332 Row 3 of A325000, col. 4 of A007318.
%Y A000332 Cf. A000292, A025036, A099174.
%K A000332 nonn,easy,nice,changed
%O A000332 0,6
%A A000332 _N. J. A. Sloane_
%E A000332 Some formulas that referred to another offset corrected by _R. J. Mathar_, Jul 07 2009
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