

A000332


Binomial coefficient binomial(n,4) = n*(n1)*(n2)*(n3)/24.
(Formerly M3853 N1578)


361



0, 0, 0, 0, 1, 5, 15, 35, 70, 126, 210, 330, 495, 715, 1001, 1365, 1820, 2380, 3060, 3876, 4845, 5985, 7315, 8855, 10626, 12650, 14950, 17550, 20475, 23751, 27405, 31465, 35960, 40920, 46376, 52360, 58905, 66045, 73815, 82251, 91390, 101270, 111930, 123410
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,6


COMMENTS

Number of intersection points of diagonals of convex ngon where no more than two diagonals intersect at any point in the interior.
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]
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
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.)
Figurate numbers based on the 4dimensional regular convex polytope called the regular 4simplex, pentachoron, 5cell, pentatope or 4hypertetrahedron with Schlaefli symbol {3,3,3}. a(n)=((n*(n1)*(n2)*(n3))/4!).  Michael J. Welch (mjw1(AT)ntlworld.com), Apr 01 2004, R. J. Mathar, Jul 07 2009
Maximal number of crossings that can be created by connecting n vertices with straight lines.  Cameron RedsellMontgomerie (credsell(AT)uoguelph.ca), Jan 30 2007
If X is an nset and Y a fixed (n1)subset of X then a(n) is equal to the number of 4subsets of X intersecting Y.  Milan Janjic, Aug 15 2007
Product of four consecutive numbers divided by 24.  Artur Jasinski, Dec 02 2007
The only prime in this sequence is 5.  Artur Jasinski, Dec 02 2007
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 eightcharacter string has 5 solutions, nine has 15, ten has 35 and so on, congruent to A000332.  Gil Broussard, Mar 19 2008
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
Nonzero terms = row sums of triangle A158824.  Gary W. Adamson, Mar 28 2009
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
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
For n>=1, a(n) is the number of ndigit numbers the binary expansion of which contains two runs of 0's.  Vladimir Shevelev, Jul 30 2010
For n>0, a(n) is the number of crossing set partitions of {1,2,..,n} into n2 blocks.  Peter Luschny, Apr 29 2011
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
For n > 3, a(n) is the hyperWiener index of the path graph on n2 vertices.  Emeric Deutsch, Feb 15 2012
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
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.
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
Does not satisfy Benford's law [Ross, 2012]  N. J. A. Sloane, Feb 12 2017
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
From Mircea Dan Rus, Aug 26 2020: (Start)
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]
_
__
___
____
____
(End)
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 length2 words is 15: aa,ab,ac,ad,ae,bb,bc,bd,be,cc,cd,ce,dd,de,ee.  Jim Nastos, Jan 18 2021
From Tom Copeland, Jun 07 2021: (Start)
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.
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^{nk}, 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.
See A099174 and A000292 for analogous relationships for the third and fourth diagonals of the Pascal matrix. (End)


REFERENCES

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.
A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 196.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 74, Problem 8.
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.
J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Charles W. Trigg: Mathematical Quickies. New York: Dover Publications, Inc., 1985, p. 53, #191


LINKS

Franklin T. AdamsWatters, Table of n, a(n) for n = 0..1002
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].
Brandy Amanda Barnette, Counting Convex Sets on Products of Totally Ordered Sets, Masters Theses & Specialist Projects, Paper 1484, 2015.
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Ömür Deveci and Anthony G. Shannon, Some aspects of Neyman triangles and Delannoy arrays, Mathematica Montisnigri (2021) Vol. L, 3643.
Paul Erdős, Norbert Kaufman, R. H. Koch and Arthur Rosenthal, E750 (Interior diagonal points), Amer. Math. Monthly, 54 (Jun, 1947), p. 344.
Th. Grüner, A. Kerber, R. Laue and M. Meringer, Mathematics for Combinatorial Chemistry.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 254.
Milan Janjic, Two Enumerative Functions.
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2002), 6575.
Tim McDevitt and Kathryn Sutcliffe, A New Look at an Old Triangle Counting Problem. The Mathematics Teacher. Vol. 110, No. 6 (February 2017), pp. 470474.
Rajesh Kumar Mohapatra and TzungPei Hong, On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences, Mathematics (2022) Vol. 10, No. 7, 1161.
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some WellKnown Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Les Reid, Counting Triangles in an Array.
Les Reid, Counting Triangles in an Array. [Cached copy]
Luis Manuel Rivera, Integer sequences and kcommuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
Kenneth A. Ross, First Digits of Squares and Cubes, Math. Mag. 85 (2012) 3642. doi:10.4169/math.mag.85.1.36.
Kirill S. Shardakov and Vladimir P. Bubnov, Stochastic Model of a HighLoaded 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), 2934.
Eric Weisstein's World of Mathematics, Composition.
Eric Weisstein's World of Mathematics, Pentatope Number.
Eric Weisstein's World of Mathematics, Pentatope.
A. F. Y. Zhao, Pattern Popularity in Multiply Restricted Permutations, Journal of Integer Sequences, 17 (2014), #14.10.3.
Index to sequences related to pyramidal numbers
Index entries for linear recurrences with constant coefficients, signature (5,10,10,5,1).
Index entries for sequences related to Benford's law


FORMULA

a(n) = n*(n1)*(n2)*(n3)/24.
G.f.: x^4/(1x)^5.  Simon Plouffe in his 1992 dissertation
a(n) = n*a(n1)/(n4).  Benoit Cloitre, Apr 26 2003, R. J. Mathar, Jul 07 2009
a(n) = Sum_{k=1..n3} Sum_{i=1..k} i*(i+1)/2.  Benoit Cloitre, Jun 15 2003
Convolution of natural numbers {1, 2, 3, 4, ...} and A000217, the triangular numbers {1, 3, 6, 10, ...}.  Jon Perry, Jun 25 2003
a(n) = A110555(n+1,4).  Reinhard Zumkeller, Jul 27 2005
a(n+1) = ((n^5(n1)^5)  (n^3(n1)^3))/24  (n^5(n1)^51)/30; a(n) = A006322(n2)A006325(n1).  Xavier Acloque, Oct 20 2003; R. J. Mathar, Jul 07 2009
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) = Bth Agonal pyramid number = ((A2)*B^3 + 3*B^2  (A5)*B)/6; For all positive integers i and the pentagonal number function P(x) = x*(3*x1)/2: a(3*i2) = P(P(i)) and a(3*i1) = P(P(i) + i); 1 + 24*a(n) = (n^2 + 3*n + 1)^2.  Jonathan Vos Post, Nov 15 2004
First differences of A000389(n).  Alexander Adamchuk, Dec 19 2004
For n > 3, the sum of the first n2 tetrahedral numbers (A000292).  Martin Steven McCormick (mathseq(AT)wazer.net), Apr 06 2005 [Corrected by Doug Bell, Jun 25 2017]
Starting (1, 5, 15, 35, ...), = binomial transform of [1, 4, 6, 4, 1, 0, 0, 0, ...].  Gary W. Adamson, Dec 28 2007
Sum_{n>=4} 1/a(n) = 4/3, from the Taylor expansion of (1x)^3*log(1x) in the limit x>1.  R. J. Mathar, Jan 27 2009
A034263(n) = (n+1)*a(n+4)  Sum_{i=0..n+3} a(i). Also A132458(n) = a(n)^2  a(n1)^2 for n>0.  Bruno Berselli, Dec 29 2010
a(n) = 5*a(n1)  10*a(n2) + 10*a(n3)  5*a(n4) + a(n5); a(0)=0, a(1)=0, a(2)=0, a(3)=0, a(4)=1.  Harvey P. Dale, Aug 22 2011
a(n) = (binomial(n1,2)^2  binomial(n1,2))/6.  Gary Detlefs, Nov 20 2011
a(n) = Sum_{k=1..n2} Sum_{i=1..k} i*(nk2).  Wesley Ivan Hurt, Sep 25 2013
a(n) = (A000217(A000217(n2)  1))/3 = ((((n2)^2 + (n2))/2)^2  (((n2)^2 + (n2))/2))/(2*3).  Raphie Frank, Jan 16 2014
Sum_{n>=0} a(n)/n! = e/24. Sum_{n>=3} a(n)/(n3)! = 73*e/24. See A067764 regarding the second ratio.  Richard R. Forberg, Dec 26 2013
Sum_{n>=4} (1)^(n+1)/a(n) = 32*log(2)  64/3 = A242023 = 0.847376444589... .  Richard R. Forberg, Aug 11 2014
4/(Sum_{n>=m} 1/a(n)) = A027480(m3), for m>=4.  Richard R. Forberg, Aug 12 2014
E.g.f.: x^4*exp(x)/24.  Robert Israel, Nov 23 2014
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.
a(n) = A080852(1,n4).  R. J. Mathar, Jul 28 2016
From Gary W. Adamson, Feb 06 2017: (Start)
G.f.: Starting (1, 5, 14, ...), x/(1x)^5 can be written
as (x * r(x) * r(x^2) * r(x^4) * r(x^8) * ...) where r(x) = (1+x)^5;
as (x * r(x) * r(x^3) * r(x^9) * r(x^27) * ...) where r(x) = (1+x+x^2)^5;
as (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). (End)
From Robert A. Russell, Jan 22 2020: (Start)
a(n) = A006008(n)  a(n+3) = (A006008(n)  A006003(n)) / 2 = a(n+3)  A006003(n).
a(n+3) = A006008(n)  a(n) = (A006008(n) + A006003(n)) / 2 = a(n) + A006003(n).
a(n) = A007318(n,4).
a(n+3) = A325000(3,n). (End)
Product_{n>=5} (1  1/a(n)) = cosh(sqrt(15)*Pi/2)/(100*Pi).  Amiram Eldar, Jan 21 2021


EXAMPLE

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


MAPLE

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


MATHEMATICA

Table[ Binomial[n, 4], {n, 0, 45} ] (* corrected by Harvey P. Dale, Aug 22 2011 *)
Table[(n4)(n3)(n2)(n1)/24, {n, 100}] (* Artur Jasinski, Dec 02 2007 *)
LinearRecurrence[{5, 10, 10, 5, 1}, {0, 0, 0, 0, 1}, 45] (* Harvey P. Dale, Aug 22 2011 *)
CoefficientList[Series[x^4 / (1  x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Nov 23 2014 *)


PROG

(PARI) a(n)=binomial(n, 4);
(Magma) [Binomial(n, 4): n in [0..50]]; // Vincenzo Librandi, Nov 23 2014
(GAP) A000332 := List([1..10^2], n > Binomial(n, 4)); # Muniru A Asiru, Oct 16 2017
(Python 3)
# Starts at a(3), i.e. computes n*(n+1)*(n+2)*(n+3)/24
# which is more in line with A000217 and A000292.
def A000332():
x, y, z, u = 1, 1, 1, 1
yield 0
while True:
yield x
x, y, z, u = x + y + z + u + 1, y + z + u + 1, z + u + 1, u + 1
a = A000332(); print([next(a) for i in range(41)]) # Peter Luschny, Aug 03 2019
(Python)
print([n*(n1)*(n2)*(n3)//24 for n in range(50)])
# Gennady Eremin, Feb 06 2022


CROSSREFS

Cf. A053134, A053126, A000389, A000579A000582, A075733, A006322, A006325.
Cf. also A000583, A014820, A092181, A092182, A092183.
Cf. A000217, A000292, A007318 (column k = 4).
Cf. A158824.
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).
Cf. A104712 (third column, k=4).
See A269747 for a 3D analog.
Cf. A006008 (oriented), A006003 (achiral) tetrahedron colorings.
Row 3 of A325000, col. 4 of A007318.
Cf. A000292, A025036, A099174.
Sequence in context: A138779 A341184 A090580 * A342213 A140227 A264925
Adjacent sequences: A000329 A000330 A000331 * A000333 A000334 A000335


KEYWORD

nonn,easy,nice


AUTHOR

N. J. A. Sloane


EXTENSIONS

Some formulas that referred to another offset corrected by R. J. Mathar, Jul 07 2009


STATUS

approved



