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A000389 Binomial coefficients C(n,5).

%I A000389 M4142 N1719 #259 Jul 09 2025 07:39:52
%S A000389 0,0,0,0,0,1,6,21,56,126,252,462,792,1287,2002,3003,4368,6188,8568,
%T A000389 11628,15504,20349,26334,33649,42504,53130,65780,80730,98280,118755,
%U A000389 142506,169911,201376,237336,278256,324632,376992,435897,501942,575757,658008,749398
%N A000389 Binomial coefficients C(n,5).
%C A000389 a(n+4) is the number of inequivalent ways of coloring the vertices of a regular 4-dimensional simplex with n colors, under the full symmetric group S_5 of order 120, with cycle index (x1^5 + 10*x1^3*x2 + 20*x1^2*x3 + 15*x1*x2^2 + 30*x1*x4 + 20*x2*x3 + 24*x5)/120.
%C A000389 Figurate numbers based on 5-dimensional regular simplex. According to Hyun Kwang Kim, it appears that every nonnegative integer can be represented as the sum of g = 10 of these 5-simplex(n) numbers (compared with g=3 for triangular numbers, g=5 for tetrahedral numbers and g=8 for pentatope numbers). - _Jonathan Vos Post_, Nov 28 2004
%C A000389 The convolution of the nonnegative integers (A001477) with the tetrahedral numbers (A000292), which are the convolution of the nonnegative integers with themselves (making appropriate allowances for offsets of all sequences). - _Graeme McRae_, Jun 07 2006
%C A000389 a(n) is the number of terms in the expansion of (a_1 + a_2 + a_3 + a_4 + a_5 + a_6)^n. - _Sergio Falcon_, Feb 12 2007
%C A000389 Product of five consecutive numbers divided by 120. - _Artur Jasinski_, Dec 02 2007
%C A000389 Equals binomial transform of [1, 5, 10, 10, 5, 1, 0, 0, 0, ...]. - _Gary W. Adamson_, Feb 02 2009
%C A000389 Equals INVERTi transform of A099242 (1, 7, 34, 153, 686, 3088, ...). - _Gary W. Adamson_, Feb 02 2009
%C A000389 For a team with n basketball players (n>=5), this sequence is the number of possible starting lineups of 5 players, without regard to the positions (center, forward, guard) of the players. - _Mohammad K. Azarian_, Sep 10 2009
%C A000389 a(n) is the number of different patterns, regardless of order, when throwing (n-5) 6-sided dice. For example, one die can display the 6 numbers 1, 2, ..., 6; two dice can display the 21 digit-pairs 11, 12, ..., 56, 66. - _Ian Duff_, Nov 16 2009
%C A000389 Sum of the first n pentatope numbers (1, 5, 15, 35, 70, 126, 210, ...), see A000332. - _Paul Muljadi_, Dec 16 2009
%C A000389 Sum_{n>=0} a(n)/n! = e/120. Sum_{n>=4} a(n)/(n-4)! = 501*e/120. See A067764 regarding the second ratio. - _Richard R. Forberg_, Dec 26 2013
%C A000389 For a set of integers {1,2,...,n}, a(n) is the sum of the 2 smallest elements of each subset with 4 elements, which is 3*C(n+1,5) (for n>=4), hence a(n) = 3*C(n+1,5) = 3*A000389(n+1). - _Serhat Bulut_, Mar 11 2015
%C A000389 a(n) = fallfac(n,5)/5! is also the number of independent components of an antisymmetric tensor of rank 5 and dimension n >= 1. Here fallfac is the falling factorial. - _Wolfdieter Lang_, Dec 10 2015
%C A000389 Number of compositions (ordered partitions) of n+1 into exactly 6 parts. - _Juergen Will_, Jan 02 2016
%C A000389 Number of weak compositions (ordered weak partitions) of n-5 into exactly 6 parts. - _Juergen Will_, Jan 02 2016
%C A000389 a(n+3) could be the general number of all geodetic graphs of diameter n>=2 homeomorphic to the Petersen Graph. - _Carlos Enrique Frasser_, May 24 2018
%C A000389 From _Robert A. Russell_, Dec 24 2020: (Start)
%C A000389 a(n) is the number of chiral pairs of colorings of the 5 tetrahedral facets (or vertices) of the regular 4-D simplex (5-cell, pentachoron, Schläfli symbol {3,3,3}) using subsets of a set of n colors. Each member of a chiral pair is a reflection but not a rotation of the other.
%C A000389 a(n+4) is the number of unoriented colorings of the 5 tetrahedral facets of the regular 4-D simplex (5-cell, pentachoron) using subsets of a set of n colors. Each chiral pair is counted as one when enumerating unoriented arrangements. (End)
%C A000389 For integer m and positive integer r >= 4, 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) = (4 - r*m)/2 in the complex plane. - _Peter Bala_, Jun 02 2024
%D A000389 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 A000389 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 196.
%D A000389 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 A000389 Gupta, Hansraj; Partitions of j-partite numbers into twelve or a smaller number of parts. Collection of articles dedicated to Professor P. L. Bhatnagar on his sixtieth birthday. Math. Student 40 (1972), 401-441 (1974).
%D A000389 J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
%D A000389 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000389 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000389 T. D. Noe, <a href="/A000389/b000389.txt">Table of n, a(n) for n = 0..1000</a>
%H A000389 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H A000389 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 A000389 Serhat Bulut, <a href="http://www.matematikproje.com/dosyalar/7e1cdSubset_smallest_elements_Sum.pdf">Subset Sum Problem</a>, 2015.
%H A000389 P. 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 A000389 C. E. Frasser and G. N. Vostrov, <a href="https://arxiv.org/abs/1611.01873">Geodetic Graphs Homeomorphic to a Given Geodetic Graph</a>, arXiv:1611.01873 [cs.DM], 2016. [p. 27]
%H A000389 H. Gupta, <a href="/A001840/a001840.pdf">Partitions of j-partite numbers into twelve or a smaller number of parts</a>, Math. Student 40 (1972), 401-441 (1974). [Annotated scanned copy]
%H A000389 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=255">Encyclopedia of Combinatorial Structures 255</a>
%H A000389 H. K. Kim, <a href="http://dx.doi.org/10.1090/S0002-9939-02-06710-2">On Regular Polytope Numbers</a>, Proc. Amer.Math. Soc. 131 (2003), 65-75.
%H A000389 Iva Kodrnja and Helena Koncul, <a href="https://hrcak.srce.hr/file/470068">Polynomials vanishing on a basis of S_m(Gamma_0(N))</a>, Glasnik Matematički (2024) Vol. 59, No. 79, 313-325. See p. 324.
%H A000389 Feihu Liu, Guoce Xin, and Chen Zhang, <a href="https://arxiv.org/abs/2412.18744">Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS</a>, arXiv:2412.18744 [math.CO], 2024. See pp. 13, 15.
%H A000389 P. A. MacMahon, <a href="http://www.jstor.org/stable/90632">Memoir on the Theory of the Compositions of Numbers</a>, Phil. Trans. Royal Soc. London A, 184 (1893), 835-901. - _Juergen Will_, Jan 02 2016
%H A000389 Ângela Mestre and José Agapito, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Mestre/mestre2.html">Square Matrices Generated by Sequences of Riordan Arrays</a>, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
%H A000389 Rajesh Kumar Mohapatra and Tzung-Pei Hong, <a href="https://doi.org/10.3390/math10071161">On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences</a>, Mathematics (2022) Vol. 10, No. 7, 1161.
%H A000389 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 A000389 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; arXiv:0911.4975 [math.NT], 2009.
%H A000389 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A000389 J. V. Post, <a href="http://www.magicdragon.com/poly.html">Table of Polytope Numbers, Sorted, Through 1,000,000</a>.
%H A000389 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Composition.html">Composition.</a>
%H A000389 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 A000389 <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>
%H A000389 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F A000389 G.f.: x^5/(1-x)^6.
%F A000389 a(n) = n*(n-1)*(n-2)*(n-3)*(n-4)/120.
%F A000389 a(n) = (n^5-10*n^4+35*n^3-50*n^2+24*n)/120. (Replace all x_i's in the cycle index with n.)
%F A000389 a(n+2) = Sum_{i+j+k=n} i*j*k. - _Benoit Cloitre_, Nov 01 2002
%F A000389 Convolution of triangular numbers (A000217) with themselves.
%F A000389 Partial sums of A000332. - _Alexander Adamchuk_, Dec 19 2004
%F A000389 a(n) = -A110555(n+1,5). - _Reinhard Zumkeller_, Jul 27 2005
%F A000389 a(n+3) = (1/2!)*(d^2/dx^2)S(n,x)|_{x=2}, n>=2, one half of second derivative of Chebyshev S-polynomials evaluated at x=2. See A049310. - _Wolfdieter Lang_, Apr 04 2007
%F A000389 a(n) = A052787(n+5)/120. - _Zerinvary Lajos_, Apr 26 2007
%F A000389 Sum_{n>=5} 1/a(n) = 5/4. - _R. J. Mathar_, Jan 27 2009
%F A000389 For n>4, a(n) = 1/(Integral_{x=0..Pi/2} 10*(sin(x))^(2*n-9)*(cos(x))^9). - _Francesco Daddi_, Aug 02 2011
%F A000389 Sum_{n>=5} (-1)^(n + 1)/a(n) = 80*log(2) - 655/12 = 0.8684411114... - _Richard R. Forberg_, Aug 11 2014
%F A000389 a(n) = -a(4-n) for all n in Z. - _Michael Somos_, Oct 07 2014
%F A000389 0 = a(n)*(+a(n+1) + 4*a(n+2)) + a(n+1)*(-6*a(n+1) + a(n+2)) for all n in Z. - _Michael Somos_, Oct 07 2014
%F A000389 a(n) = 3*C(n+1, 5) = 3*A000389(n+1). - _Serhat Bulut_, Mar 11 2015
%F A000389 From _Ilya Gutkovskiy_, Jul 23 2016: (Start)
%F A000389 E.g.f.: x^5*exp(x)/120.
%F A000389 Inverse binomial transform of A054849. (End)
%F A000389 From _Robert A. Russell_, Dec 24 2020: (Start)
%F A000389 a(n) = A337895(n) - a(n+4) = (A337895(n) - A132366(n-1)) / 2 = a(n+4) - A132366(n-1).
%F A000389 a(n+4) = A337895(n) - a(n) = (A337895(n) + A132366(n-1)) / 2 = a(n) + A132366(n-1).
%F A000389 a(n+4) = 1*C(n,1) + 4*C(n,2) + 6*C(n,3) + 4*C(n,4) + 1*C(n,5), where the coefficient of C(n,k) is the number of unoriented pentachoron colorings using exactly k colors. (End)
%e A000389 G.f. = x^5 + 6*x^6 + 21*x^7 + 56*x^8 + 126*x^9 + 252*x^10 + 462*x^11 + ...
%e A000389 For A={1,2,3,4}, the only subset with 4 elements is {1,2,3,4}; sum of 2 minimum elements of this subset: a(4) = 1+2 = 3 = 3*C(4+1,5).
%e A000389 For A={1,2,3,4,5}, the subsets with 4 elements are {1,2,3,4}, {1,2,3,5}, {1,2,4,5}, {1,3,4,5}, {2,3,4,5}; sum of 2 smallest elements of each subset: a(5) = (1+2)+(1+2)+(1+2)+(1+3)+(2+3) = 18 = 3*C(5+1,5). - _Serhat Bulut_, Mar 11 2015
%e A000389 a(6) = 6 from the six independent components of an antisymmetric tensor A of rank 5 and dimension 6: A(1,2,3,4,5), A(1,2,3,4,6), A(1,2,3,5,6), A(1,2,4,5,6), A(1,3,4,5,6), A(2,3,4,5,6). See the Dec 10 2015 comment. - _Wolfdieter Lang_, Dec 10 2015
%p A000389 f:=n->(1/120)*(n^5-10*n^4+35*n^3-50*n^2+24*n): seq(f(n), n=0..60);
%p A000389 ZL := [S, {S=Prod(B,B,B,B,B,B), B=Set(Z, 1 <= card)}, unlabeled]: seq(combstruct[count](ZL, size=n+1), n=0..42); # _Zerinvary Lajos_, Mar 13 2007
%p A000389 A000389:=1/(z-1)**6; # _Simon Plouffe_, 1992 dissertation
%t A000389 Table[Binomial[n, 5], {n, 5, 50}] (* _Stefan Steinerberger_, Apr 02 2006 *)
%t A000389 CoefficientList[Series[x^5 / (1 - x)^6, {x, 0, 40}], x] (* _Vincenzo Librandi_, Mar 12 2015 *)
%t A000389 LinearRecurrence[{6,-15,20,-15,6,-1},{0,0,0,0,0,1},50] (* _Harvey P. Dale_, Jul 17 2016 *)
%o A000389 (PARI) (conv(u,v)=local(w); w=vector(length(u),i,sum(j=1,i,u[j]*v[i+1-j])); w);
%o A000389 (t(n)=n*(n+1)/2); u=vector(10,i,t(i)); conv(u,u)
%o A000389 (Haskell)
%o A000389 a000389 n = a000389_list !! n
%o A000389 a000389_list = 0 : 0 : f [] a000217_list where
%o A000389    f xs (t:ts) = (sum $ zipWith (*) xs a000217_list) : f (t:xs) ts
%o A000389 -- _Reinhard Zumkeller_, Mar 03 2015, Apr 13 2012
%o A000389 (Magma) [Binomial(n, 5): n in [0..40]]; // _Vincenzo Librandi_, Mar 12 2015
%Y A000389 Cf. A002299, A053127, A000332, A000579, A000580, A000581, A000582.
%Y A000389 Cf. A000217, A005583, A051747, A000292.
%Y A000389 Cf. A099242. - _Gary W. Adamson_, Feb 02 2009
%Y A000389 Cf. A242023. A104712 (fourth column, k=5).
%Y A000389 Cf. A001477, A049310, A052787, A067764, A110555, A277935.
%Y A000389 5-cell colorings: A337895 (oriented), A132366(n-1) (achiral).
%Y A000389 Unoriented colorings: A063843 (5-cell edges, faces), A128767 (8-cell vertices, 16-cell facets), A337957 (16-cell vertices, 8-cell facets), A338949 (24-cell), A338965 (600-cell vertices, 120-cell facets).
%Y A000389 Chiral colorings: A331352 (5-cell edges, faces), A337954 (8-cell vertices, 16-cell facets), A234249 (16-cell vertices, 8-cell facets), A338950 (24-cell), A338966 (600-cell vertices, 120-cell facets).
%K A000389 nonn,easy,nice
%O A000389 0,7
%A A000389 _N. J. A. Sloane_
%E A000389 Corrected formulas that had been based on other offsets. - _R. J. Mathar_, Jun 16 2009
%E A000389 I changed the offset to 0. This will require some further adjustments to the formulas. - _N. J. A. Sloane_, Aug 01 2010