cp's OEIS Frontend

A000580 a(n) = binomial coefficient C(n,7).

%I A000580 M4517 N1911 #162 Jul 09 2025 07:39:23
%S A000580 1,8,36,120,330,792,1716,3432,6435,11440,19448,31824,50388,77520,
%T A000580 116280,170544,245157,346104,480700,657800,888030,1184040,1560780,
%U A000580 2035800,2629575,3365856,4272048,5379616,6724520,8347680,10295472
%N A000580 a(n) = binomial coefficient C(n,7).
%C A000580 Figurate numbers based on 7-dimensional regular simplex. According to Hyun Kwang Kim, it appears that every nonnegative integer can be represented as the sum of g = 15 of these numbers. - _Jonathan Vos Post_, Nov 28 2004
%C A000580 a(n) is the number of terms in the expansion of (Sum_{i=1..8} a_i)^n. - _Sergio Falcon_, Feb 12 2007
%C A000580 Product of seven consecutive numbers divided by 7!. - _Artur Jasinski_, Dec 02 2007
%C A000580 In this sequence there are no primes. - _Artur Jasinski_, Dec 02 2007
%C A000580 For a set of integers {1,2,...,n}, a(n) is the sum of the 2 smallest elements of each subset with 6 elements, which is 3*C(n+1,7) (for n>=6), hence a(n) = 3*C(n+1,7) = 3*A000580(n+1). - _Serhat Bulut_, Mar 13 2015
%C A000580 Partial sums of A000579. In general, the iterated sums S(m, n) = Sum_{j=1..n} S(m-1, j) with input S(1, n) = A000217(n) = 1 + 2 + ... + n are S(m, n) = risefac(n, m+1)/(m+1)! = binomial(n+m, m+1) = Sum_{k = 1..n} risefac(k, m)/m!, with the rising factorials risefac(x, m):= Product_{j=0..m-1} (x+j), for m >= 1. Such iterated sums of arithmetic progressions have been considered by Narayana Pandit (see The MacTutor History of Mathematics archive link, and the Gottwald et al. reference, p. 338, where the name Narayana Daivajna is also used). - _Wolfdieter Lang_, Mar 20 2015
%C A000580 a(n) = fallfac(n,7)/7! = binomial(n, 7) is also the number of independent components of an antisymmetric tensor of rank 7 and dimension n >= 7 (for n=1..6 this becomes 0). Here fallfac is the falling factorial. - _Wolfdieter Lang_, Dec 10 2015
%C A000580 From _Juergen Will_, Jan 02 2016: (Start)
%C A000580 Number of compositions (ordered partitions) of n+1 into exactly 8 parts.
%C A000580 Number of weak compositions (ordered weak partitions) of n-7 into exactly 8 parts. (End)
%D A000580 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 A000580 Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 196.
%D A000580 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 A000580 S. Gottwald, H.-J. Ilgauds and K.-H. Schlote (eds.), Lexikon bedeutender Mathematiker (in German), Bibliographisches Institut Leipzig, 1990.
%D A000580 J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
%D A000580 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000580 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000580 T. D. Noe, <a href="/A000580/b000580.txt">Table of n, a(n) for n = 7..1000</a>
%H A000580 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 A000580 Serhat Bulut and Oktay Erkan Temizkan, <a href="http://matematikproje.com/dosyalar/7e1cdSubset_smallest_elements_Sum.pdf">Subset Sum Problem</a>, 2015
%H A000580 Peter 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 A000580 Philippe A. J. G. Chevalier, <a href="https://www.researchgate.net/publication/236594822_On_the_discrete_geometry_of_physical_quantities">On the discrete geometry of physical quantities</a>, Preprint, 2012.
%H A000580 Philippe A. J. G. Chevalier, <a href="https://www.researchgate.net/publication/262067273_A_table_of_Mendeleev_for_physical_quantities">A "table of Mendeleev" for physical quantities?</a>, Slides from a talk, May 14 2014, Leuven, Belgium.
%H A000580 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=257">Encyclopedia of Combinatorial Structures 257</a>.
%H A000580 Milan Janjic, <a href="https://old.pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>.
%H A000580 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., Vol. 131, No. 1 (2002), pp. 65-75.
%H A000580 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 A000580 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 A000580 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 A000580 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 A000580 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 A000580 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A000580 Jonathan Vos Post, <a href="https://web.archive.org/web/20200219170305/http://www.magicdragon.com/poly.html">Table of Polytope Numbers, Sorted, Through 1,000,000</a>.
%H A000580 The MacTutor History of Mathematics archive, <a href="http://www-history.mcs.st-and.ac.uk/Mathematicians/Narayana.html">Narayana Pandit</a>.
%H A000580 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Composition.html">Composition</a>.
%H A000580 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8,-1).
%F A000580 G.f.: x^7/(1-x)^8.
%F A000580 a(n) = (n^7 - 21*n^6 + 175*n^5 - 735*n^4 + 1624*n^3 - 1764*n^2 + 720*n)/5040.
%F A000580 a(n) = -A110555(n+1,7). - _Reinhard Zumkeller_, Jul 27 2005
%F A000580 Convolution of the nonnegative numbers (A001477) with the sequence A000579. Also convolution of the triangular numbers (A000217) with the sequence A000332. Also convolution of the sequence {1,1,1,1,...} (A000012) with the sequence A000579. Also self-convolution of the tetrahedral numbers (A000292). - _Sergio Falcon_, Feb 12 2007
%F A000580 a(n+4) = (1/3!)*(d^3/dx^3)S(n,x)|_{x=2}, n >= 3. One sixth of third derivative of Chebyshev S-polynomials evaluated at x=2. See A049310. - _Wolfdieter Lang_, Apr 04 2007
%F A000580 a(n) = n(n-1)(n-2)(n-3)(n-4)(n-5)(n-6)/7!. - _Artur Jasinski_, Dec 02 2007, _R. J. Mathar_, Jul 07 2009
%F A000580 a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) with a(7)=1, a(8)=8, a(9)=36, a(10)=120, a(11)=330, a(12)=792, a(13)=1716, a(14)=3432. - _Harvey P. Dale_, Nov 28 2011
%F A000580 a(n) = 3*C(n+1,7) = 3*A000580(n+1). - _Serhat Bulut_, Mar 13 2015
%F A000580 From _Wolfdieter Lang_, Mar 21 2015: (Start)
%F A000580 a(n) = A104712(n, 7), n >= 7.
%F A000580 a(n+6) = sum(A000579(j+5), j = 1..n), n >= 1. See the Mar 20 2015 comment above. (End)
%F A000580 Sum_{k >= 7} 1/a(k) = 7/6. - _Tom Edgar_, Sep 10 2015
%F A000580 Sum_{n>=7} (-1)^(n+1)/a(n) = A001787(7)*log(2) - A242091(7)/6! = 448*log(2) - 9289/30 = 0.8966035575... - _Amiram Eldar_, Dec 10 2020
%e A000580 For A={1,2,3,4,5,6,7}, subsets with 6 elements are {1,2,3,4,5,6}, {1,2,3,4,5,7}, {1,2,3,4,6,7}, {1,2,3,5,6,7}, {1,2,4,5,6,7}, {1,3,4,5,6,7}, {2,3,4,5,6,7}.
%e A000580 Sum of 2 smallest elements of each subset: a(7) = (1+2)+(1+2)+(1+2)+(1+2)+(1+2)+(1+3)+(2+3) = 24 = 3*C(7+1,7) = 3*A000580(7+1). - _Serhat Bulut_, Mar 13 2015
%p A000580 ZL := [S, {S=Prod(B,B,B,B,B,B,B,B), B=Set(Z, 1 <= card)}, unlabeled]: seq(combstruct[count](ZL, size=n), n=8..38); # _Zerinvary Lajos_, Mar 13 2007
%p A000580 A000580:=1/(z-1)**8; # _Simon Plouffe_ in his 1992 dissertation, offset 0.
%p A000580 seq(binomial(n+7,7)*1^n,n=0..30); # _Zerinvary Lajos_, Jun 23 2008
%p A000580 G(x):=x^7*exp(x): f[0]:=G(x): for n from 1 to 38 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n]/7!,n=7..37); # _Zerinvary Lajos_, Apr 05 2009
%t A000580 Table[n(n + 1)(n + 2)(n + 3)(n + 4)(n + 5)(n + 6)/7!, {n, 1, 100}] (* _Artur Jasinski_, Dec 02 2007 *)
%t A000580 Binomial[Range[7,40],7] (* or *) LinearRecurrence[ {8,-28,56,-70,56,-28,8,-1},{1,8,36,120,330,792,1716,3432},40] (* _Harvey P. Dale_, Nov 28 2011 *)
%t A000580 CoefficientList[Series[1 / (1-x)^8, {x, 0, 33}], x] (* _Vincenzo Librandi_, Mar 21 2015 *)
%o A000580 (Magma) [Binomial(n,7): n in [7..40]]; // _Vincenzo Librandi_, Mar 21 2015
%o A000580 (PARI) a(n)=binomial(n,7) \\ _Charles R Greathouse IV_, Sep 24 2015
%Y A000580 Cf. A053136, A053129, A000579, A000581, A000582, A000217, A000292, A000332, A000389, A104712, A001787, A242091.
%K A000580 nonn,easy
%O A000580 7,2
%A A000580 _N. J. A. Sloane_
%E A000580 More terms from Larry Reeves (larryr(AT)acm.org), Mar 17 2000
%E A000580 Some formulas that referred to other offsets corrected by _R. J. Mathar_, Jul 07 2009