|
%I M4142 N1719 #244 Sep 08 2022 08:44:27
%S 0,0,0,0,0,1,6,21,56,126,252,462,792,1287,2002,3003,4368,6188,8568,
%T 11628,15504,20349,26334,33649,42504,53130,65780,80730,98280,118755,
%U 142506,169911,201376,237336,278256,324632,376992,435897,501942,575757,658008,749398
%N Binomial coefficients C(n,5).
%C 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 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 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 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 Product of five consecutive numbers divided by 120. - _Artur Jasinski_, Dec 02 2007
%C Equals binomial transform of [1, 5, 10, 10, 5, 1, 0, 0, 0, ...]. - _Gary W. Adamson_, Feb 02 2009
%C Equals INVERTi transform of A099242 (1, 7, 34, 153, 686, 3088, ...). - _Gary W. Adamson_, Feb 02 2009
%C 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 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 Sum of the first n pentatope numbers (1, 5, 15, 35, 70, 126, 210, ...), see A000332. - _Paul Muljadi_, Dec 16 2009
%C 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 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 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 Number of compositions (ordered partitions) of n+1 into exactly 6 parts. - _Juergen Will_, Jan 02 2016
%C Number of weak compositions (ordered weak partitions) of n-5 into exactly 6 parts. - _Juergen Will_, Jan 02 2016
%C 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 From _Robert A. Russell_, Dec 24 2020: (Start)
%C 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 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)
%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. 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 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 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).
%H T. D. Noe, <a href="/A000389/b000389.txt">Table of n, a(n) for n = 0..1000</a>
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</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 Serhat Bulut, <a href="http://www.matematikproje.com/dosyalar/7e1cdSubset_smallest_elements_Sum.pdf">Subset Sum Problem</a>, 2015.
%H 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 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 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 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=255">Encyclopedia of Combinatorial Structures 255</a>
%H 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 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 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 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 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; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H J. V. Post, <a href="http://www.magicdragon.com/poly.html">Table of Polytope Numbers, Sorted, Through 1,000,000</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Composition.html">Composition.</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_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F G.f.: x^5/(1-x)^6.
%F a(n) = n*(n-1)*(n-2)*(n-3)*(n-4)/120.
%F 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 a(n+2) = Sum_{i+j+k=n} i*j*k. - _Benoit Cloitre_, Nov 01 2002
%F Convolution of triangular numbers (A000217) with themselves.
%F Partial sums of A000332. - _Alexander Adamchuk_, Dec 19 2004
%F a(n) = -A110555(n+1,5). - _Reinhard Zumkeller_, Jul 27 2005
%F 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 a(n) = A052787(n+5)/120. - _Zerinvary Lajos_, Apr 26 2007
%F Sum_{n>=5} 1/a(n) = 5/4. - _R. J. Mathar_, Jan 27 2009
%F 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 Sum_{n>=5} (-1)^(n + 1)/a(n) = 80*log(2) - 655/12 = 0.8684411114... - _Richard R. Forberg_, Aug 11 2014
%F a(n) = -a(4-n) for all n in Z. - _Michael Somos_, Oct 07 2014
%F 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 a(n) = 3*C(n+1, 5) = 3*A000389(n+1). - _Serhat Bulut_, Mar 11 2015
%F From _Ilya Gutkovskiy_, Jul 23 2016: (Start)
%F E.g.f.: x^5*exp(x)/120.
%F Inverse binomial transform of A054849. (End)
%F From _Robert A. Russell_, Dec 24 2020: (Start)
%F a(n) = A337895(n) - a(n+4) = (A337895(n) - A132366(n-1)) / 2 = a(n+4) - A132366(n-1).
%F a(n+4) = A337895(n) - a(n) = (A337895(n) + A132366(n-1)) / 2 = a(n) + A132366(n-1).
%F 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.
%F (End)
%e G.f. = x^5 + 6*x^6 + 21*x^7 + 56*x^8 + 126*x^9 + 252*x^10 + 462*x^11 + ...
%e 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 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 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 f:=n->(1/120)*(n^5-10*n^4+35*n^3-50*n^2+24*n): seq(f(n), n=0..60);
%p 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:=1/(z-1)**6; # _Simon Plouffe_, 1992 dissertation
%t Table[Binomial[n, 5], {n, 5, 50}] (* _Stefan Steinerberger_, Apr 02 2006 *)
%t CoefficientList[Series[x^5 / (1 - x)^6, {x, 0, 40}], x] (* _Vincenzo Librandi_, Mar 12 2015 *)
%t LinearRecurrence[{6,-15,20,-15,6,-1},{0,0,0,0,0,1},50] (* _Harvey P. Dale_, Jul 17 2016 *)
%o (PARI) (conv(u,v)=local(w); w=vector(length(u),i,sum(j=1,i,u[j]*v[i+1-j])); w);
%o (t(n)=n*(n+1)/2); u=vector(10,i,t(i)); conv(u,u)
%o (Haskell)
%o a000389 n = a000389_list !! n
%o a000389_list = 0 : 0 : f [] a000217_list where
%o f xs (t:ts) = (sum $ zipWith (*) xs a000217_list) : f (t:xs) ts
%o -- _Reinhard Zumkeller_, Mar 03 2015, Apr 13 2012
%o (Magma) [Binomial(n, 5): n in [0..40]]; // _Vincenzo Librandi_, Mar 12 2015
%Y Cf. A002299, A053127, A000332, A000579, A000580, A000581, A000582.
%Y Cf. A000217, A005583, A051747, A000292.
%Y Cf. A099242. - _Gary W. Adamson_, Feb 02 2009
%Y Cf. A242023. A104712 (fourth column, k=5).
%Y Cf. A001477, A049310, A052787, A067764, A110555, A277935.
%Y 5-cell colorings: A337895 (oriented), A132366(n-1) (achiral).
%Y 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 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 nonn,easy,nice
%O 0,7
%A _N. J. A. Sloane_
%E Corrected formulas that had been based on other offsets. - _R. J. Mathar_, Jun 16 2009
%E I changed the offset to 0. This will require some further adjustments to the formulas. - _N. J. A. Sloane_, Aug 01 2010
|
