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 A000579 Figurate numbers or binomial coefficients C(n,6). (Formerly M4390 N1847) 98

%I M4390 N1847

%S 0,0,0,0,0,0,1,7,28,84,210,462,924,1716,3003,5005,8008,12376,18564,

%T 27132,38760,54264,74613,100947,134596,177100,230230,296010,376740,

%U 475020,593775,736281,906192,1107568,1344904,1623160,1947792,2324784,2760681,3262623

%N Figurate numbers or binomial coefficients C(n,6).

%C Number of triangles (all of whose vertices lie inside the circle) formed when n points in general position on a circle are joined by straight lines - Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), May 25 2000

%C Figurate numbers based on 6-dimensional regular simplex. According to Hyun Kwang Kim, it appears that every nonnegative integer can be represented as the sum of g = 13 of these numbers. - _Jonathan Vos Post_, Nov 28 2004

%C a(n) = A110555(n+1,6). - _Reinhard Zumkeller_, Jul 27 2005

%C a(n) is the number of terms in the expansion of (a_1 + a_2 + a_3 + a_4 + a_5 + a_6 + a_7)^n. - _Sergio Falcon_, Feb 12 2007

%C Only prime in this sequence is 7. - _Artur Jasinski_, Dec 02 2007

%C 6-dimensional triangular numbers, sixth partial sums of binomial transform of [1, 0, 0, 0, ...]. - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009, _R. J. Mathar_, Jul 07 2009

%C The number of n-digit numbers the binary expansion of which contains 3 runs of 0's. Generally, the number of n-digit numbers with k runs of 0's is Sum_{i = k..n-k} binomial(i-1, k-1)*binomial(n-i, k). - _Vladimir Shevelev_, Jul 30 2010

%C The dimension of the space spanned by a 6-form that couples to M5-brane worldsheets wrapping 6-cycles inside tori (ref. Green,Miller,Vanhove eq. 3.10). - _Stephen Crowley_, Jan 09 2012

%C Sum(n >= 0, a(n)/n! ) = e/720. Sum(n >= 5, a(n)/(n-5)! ) = 4051*e/720. See A067653 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 5 elements, which is 3*C(n+1,6) (for n>=5), hence a(n) = 3*C(n+1,6) = 3*A000579(n+1). - _Serhat Bulut_, Oktay Erkan Temizkan, Mar 13 2015

%C a(n) = fallfac(n, 6)/6! is also the number of independent components of an antisymmetric tensor of rank 6 and dimension n >= 1. Here fallfac is the falling factorial. - _Wolfdieter Lang_, Dec 10 2015

%C Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative integer lattice point of the orbit, when the cardinality of the orbit is equal to 645120. - _Philippe A.J.G. Chevalier_, Dec 28 2015

%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 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).

%D Charles W. Trigg: Mathematical Quickies. New York: Dover Publications, Inc., 1985, p. 11, #32

%H T. D. Noe, <a href="/A000579/b000579.txt">Table of n, a(n) for n = 0..1000</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, Oktay Erkan Temizkan, <a href="http://web.archive.org/web/20160708101054/http://matematikproje.com/dosyalar/7e1cdSubset_smallest_elements_Sum.pdf">Subset Sum Problem</a>

%H Peter J. Cameron, <a href="https://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 Philippe A. J. G. Chevalier, <a href="http://hdl.handle.net/1854/LU-8049761">On the discrete geometry of physical quantities</a>, Preprint, 2012.

%H Philippe A. J. G. Chevalier, <a href="https://www.researchgate.net/profile/Philippe_Chevalier2/publication/260598331_On_a_Mathematical_Method_for_Discovering_Relations_Between_Physical_Quantities_a_Photonics_Case_Study/links/00b7d531be7b837626000000.pdf">On a Mathematical Method for Discovering Relations Between Physical Quantities: a Photonics Case Study</a>, Slides from a talk presented at ICOL2014.

%H Philippe A. J. G. Chevalier, <a href="http://www.researchgate.net/profile/Philippe_Chevalier2/publication/262067273_A_table_of_Mendeleev_for_physical_quantities/links/0c9605368f6d191478000000.pdf">A "table of Mendeleev" for physical quantities?</a>, Slides from a talk, May 14 2014, Leuven, Belgium.

%H Philippe A. J. G. Chevalier, <a href="https://www.researchgate.net/publication/297497200">Dimensional exploration techniques for photonics</a>, Slides of a talk, 2016.

%H Michael B. Green, Stephen D. Miller, and Pierre Vanhove, <a href="http://arxiv.org/abs/1111.2983">Small representations, string instantons, and Fourier modes of Eisenstein series</a>, arXiv:1111.2983 [hep-th], 2011-2013.

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=256">Encyclopedia of Combinatorial Structures 256</a>

%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>

%H Hyun Kwang 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 Leo Moser, <a href="http://www.jstor.org/stable/3029675">Quicky 87</a>, Mathematics Magazine, 26 (March 1953), p. 226.

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992.

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">1031 Generating Functions and Conjectures</a>, Université du Québec à Montréal, 1992.

%H Jonathan Vos Post, <a href="http://www.magicdragon.com/poly.html">Table of Polytope Numbers, Sorted, Through 1,000,000</a>.

%H Hermann Stamm-Wilbrandt, <a href="https://www.ibm.com/developerworks/community/blogs/HermannSW/entry/sum_of_pascal_s_triangle_reciprocals10">Sum of Pascal's triangle reciprocals</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Composition.html">Composition</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).

%F G.f.: x^6/(1-x)^7.

%F E.g.f.: exp(x)*x^6/720.

%F a(n) = (n^6 - 15*n^5 + 85*n^4 - 225*n^3 + 274*n^2 - 120*n)/720.

%F Conjecture: a(n+3) = Sum{0 <= k, l, m <= n; k + l + m <= n} k*l*m. - _Ralf Stephan_, May 06 2005

%F Convolution of the nonnegative numbers (A001477) with the hexagonal numbers (A000389). Also convolution of the triangular numbers (A000217) with the tetrahedral numbers (A000292). - _Sergio Falcon_, Feb 12 2007

%F a(n) = n(n - 1)(n - 2)(n - 3)(n - 4)(n - 5)/720. - _Artur Jasinski_, Dec 02 2007, _R. J. Mathar_, Jul 07 2009

%F Equals binomial transform of [1, 6, 15, 20, 15, 6, 1, 0, 0, 0, ...]. - _Gary W. Adamson_, Aug 02 2008

%F a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 0, a(4) = 0, a(5) = 0, a(6) = 1, a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7). - _Harvey P. Dale_, Dec 30 2012

%F Sum_{n >= 6} 1/a(n) = 6/5. - _Hermann Stamm-Wilbrandt_, Jul 13 2014

%F Sum_{n >= 6} (-1)^(n + 1)/a(n) = 192*log(2) - 661/5 = 0.8842586675... Also see A242023. - _Richard R. Forberg_, Aug 11 2014

%F a(n) = a(5-n) for all n in Z. - _Michael Somos_, Oct 07 2014

%F 0 = a(n)*(+a(n+1) +5*a(n+2)) + a(n+1)*(-7*a(n+1) +a(n+2)) for all n in Z. - _Michael Somos_, Oct 07 2014

%F a(n) = 3*C(n+1,6) = 3* A000579(n+1). - _Serhat Bulut_, Oktay Erkan Temizkan, Mar 13 2015

%F a(n) = A000292(n-5)*A000292(n-2)/20. - _R. J. Mathar_, Nov 29 2015

%e a(9) = 84 = (1, 3, 3, 1) dot (1, 6, 15, 20) = (1 + 18 + 45 + 20). - _Gary W. Adamson_, Aug 02 2008

%e G.f. = x^6 + 7*x^7 + 28*x^8 + 84*x^9 + 210*x^10 + 462*x^11 + 924*x^12 + ...

%e For A = {1,2,3,4,5,6} subsets with 5 elements are {1,2,3,4,5}, {1,2,3,4,6}, {1,2,3,5,6}, {1,2,4,5,6}, {1,3,4,5,6}, {2,3,4,5,6}. Sum of 2 smallest elements of each subset: a(6) = (1+2) + (1+2) + (1+2) + (1+2) + (1+3) + (2+3) = 21 = 3*C(6+1,6) = 3*A000579(6+1). - _Serhat Bulut_, Oktay Erkan Temizkan, Mar 13 2015

%e a(7) = 7 from the seven independent components of an antisymmetric tensor A of rank 6 and dimension 7: A(1,2,3,4,5,6), A(1,2,3,4,5,7), A(1,2,3,4,6,7), A(1,2,3,5,6,7) A(1,2,4,5,6,7), A(1,2,3,5,6,7) and A(2,3,4,5,6,7). See a Dec 10 2015 comment. - _Wolfdieter Lang_, Dec 10 2015

%p A000579 := n->binomial(n,6);

%p ZL := [S, {S=Prod(B,B,B,B,B,B,B), B=Set(Z, 1 <= card)}, unlabeled]: seq(combstruct[count](ZL, size=n), n=7..40); # _Zerinvary Lajos_, Mar 13 2007

%p A000579:=-1/(z-1)**7; # _Simon Plouffe_ in his 1992 dissertation, referring to offset 0.

%p seq(binomial(n,6),n=0..33); # _Zerinvary Lajos_, Jun 16 2008

%p G(x):=x^6*exp(x): f[0]:=G(x): for n from 1 to 39 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n]/6!,n=6..39); # _Zerinvary Lajos_, Apr 05 2009

%t Table[Binomial[n, 6], {n, 6, 50}] (* _Stefan Steinerberger_, Apr 02 2006 *)

%t Table[n(n - 1)(n - 2)(n - 3)(n - 4)(n - 5)/720, {n, 0, 100}] (* _Artur Jasinski_, Dec 02 2007 *)

%t LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,0,0,0,0,0,1},50] (* _Harvey P. Dale_, Dec 30 2012 *)

%t CoefficientList[ Series[ -7x^6/(x-1)^7,{x, 0, 35}], x]/7 (* _Robert G. Wilson v_, Jan 29 2015 *)

%o (PARI) a(n)=binomial(n,6) \\ _Charles R Greathouse IV_, Nov 20 2012

%o (MAGMA) [Binomial(n,6) : n in [0..50]]; // _Wesley Ivan Hurt_, Jul 13 2014

%o (Python)

%o A000579_list, m = [], [1, -5, 10, -10, 5, -1, 0]

%o for _ in range(10**2):

%o A000579_list.append(m[-1])

%o for i in range(6):

%o m[i+1] += m[i] # _Chai Wah Wu_, Jan 24 2016

%Y Cf. A053135, A053128, A000580 (partial sums), A000581, A000582, A000217, A000292, A000332, A000389 (first differences), A104712 (fifth column, k=6).

%K nonn,easy,nice

%O 0,8

%A _N. J. A. Sloane_

%E Some formulas that referred to other offsets corrected by _R. J. Mathar_, Jul 07 2009

%E I changed the offset to 0. This will require some further adjustments to the formulas. - _N. J. A. Sloane_, Aug 01 2010

%E Inserted Shevelev comment, further adaptations to offset - _R. J. Mathar_, Aug 03 2010

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Last modified November 20 18:55 EST 2018. Contains 317413 sequences. (Running on oeis4.)