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A000332 Binomial coefficient binomial(n,4) = n*(n-1)*(n-2)*(n-3)/24.
(Formerly M3853 N1578)
253
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 n-gon 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

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 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

a(n) = A110555(n+1,4). - Reinhard Zumkeller, Jul 27 2005

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

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

Product of four consecutive numbers divided by 24. - Artur Jasinski, Dec 02 2007

Only prime in this sequence is 5. - Artur Jasinski, Dec 02 2007

For strings consisting entirely of 0s and 1s, the number of unique arrangements of four 1s such that 1s 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

Apart from the first 4 zeros, this sequence also represents the partial sums of oblong numbers. That is, a(n)=n(n-1)(n-2)(n-3)/24. - Mohammad K. Azarian, Apr 03 2008, R. J. Mathar, Jul 07 2009

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 sum of the tetrahedral numbers from 0 to a tetrahedral number n. E.g., 0 + 1 = 1, 1 + 4 = 5, 5 + 10 = 15, 15 + 20 = 35, etc. - 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 n-digit 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 n-2 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 hyper-Wiener index of the path graph on n-2 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

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. Adams-Watters, 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].

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

Paul Erdos, Norbert Kaufman, R. H. Koch and Arthur Rosenthal, E750 (Interior diagonal points)Amer. Math. Monthly, 54 (Jun, 1947), p. 344.

Th. Gruner, 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), 65-75.

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

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Les Reid, Counting Triangles in an Array

Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081, 2014

Eric Weisstein's World of Mathematics, Composition

Eric Weisstein's World of Mathematics, Pentatope Number

Eric Weisstein's World of Mathematics, Pentatope

Index to sequences with linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

FORMULA

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

G.f.: x^4/(1-x)^5. - Jon Perry, Mar 31 2004

a(n) = n*a(n-1)/(n-4). - Benoit Cloitre, Apr 26 2003, R. J. Mathar, Jul 07 2009

a(n) = sum(k=1, n-3, 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+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

a(4n+2) = Pyr(n+4, 4n+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(3i-2) = P(P(i)) and a(3i-1) = 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

The sum of the first n tetrahedral numbers (A000292). - Martin Steven McCormick (mathseq(AT)wazer.net), Apr 06 2005

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..infinity} 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]

A034263(n)=(n+1)*a(n+4)-sum[a(i), i=0..n+3]. Also A132458(n)=a(n)^2-a(n-1)^2 for n>0. - Bruno Berselli, Dec 29 2010

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]

a(n) = (binomial(n-1,2)^2-binomial(n-1,2))/6. [Gary Detlefs, Nov 20 2011]

a(n) = sum_{k=1..n-2} ( sum_{i=1..k} i*(n-k-2) ). - Wesley Ivan Hurt, Sep 25 2013

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

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

Sum(n>=4, 1/a(n)) = 4/3. - Richard R. Forberg, Feb 10 2014

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(m-3), for m>=4. - Richard R. Forberg, Aug 12 2014

MAPLE

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

A000332:=-1/(z-1)^5; [Simon Plouffe in his 1992 dissertation, sequence starting at a(4).]

MATHEMATICA

Table[ Binomial[n, 4], {n, 0, 45} ]

Table[(n-4)(n-3)(n-2)(n-1)/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 *)

PROG

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

CROSSREFS

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

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

Partial sums of A001044.

Cf. A000217, A000292.

Cf. A158824

Sequence in context: A000743 A138779 A090580 * A140227 A049016 A139761

Adjacent sequences:  A000329 A000330 A000331 * A000333 A000334 A000335

KEYWORD

nonn,easy,nice,changed

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Larry Reeves (larryr(AT)acm.org), Mar 14 2000

Removed attribute "conjectured" from Simon Plouffe g.f. R. J. Mathar, Mar 11 2009

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

Corrected first Mathematica program - Harvey P. Dale, Aug 22 2011

Broken link to Hyun Kwang Kim's paper fixed by Felix Fröhlich, Jun 16 2014

STATUS

approved

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Last modified October 1 12:17 EDT 2014. Contains 247510 sequences.