A000580 - OEIS

%I M4517 N1911 #157 Feb 16 2025 08:32:21

%S 1,8,36,120,330,792,1716,3432,6435,11440,19448,31824,50388,77520,

%T 116280,170544,245157,346104,480700,657800,888030,1184040,1560780,

%U 2035800,2629575,3365856,4272048,5379616,6724520,8347680,10295472

%N a(n) = binomial coefficient C(n,7).

%C 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 a(n) is the number of terms in the expansion of (Sum_{i=1..8} a_i)^n. - _Sergio Falcon_, Feb 12 2007

%C Product of seven consecutive numbers divided by 7!. - _Artur Jasinski_, Dec 02 2007

%C In this sequence there are no primes. - _Artur Jasinski_, Dec 02 2007

%C 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 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 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 From _Juergen Will_, Jan 02 2016: (Start)

%C Number of compositions (ordered partitions) of n+1 into exactly 8 parts.

%C Number of weak compositions (ordered weak partitions) of n-7 into exactly 8 parts. (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 Albert 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 S. Gottwald, H.‐J. Ilgauds and K.‐H. Schlote (eds.), Lexikon bedeutender Mathematiker (in German), Bibliographisches Institut Leipzig, 1990.

%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="/A000580/b000580.txt">Table of n, a(n) for n = 7..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 and Oktay Erkan Temizkan, <a href="http://matematikproje.com/dosyalar/7e1cdSubset_smallest_elements_Sum.pdf">Subset Sum Problem</a>, 2015

%H 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 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 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 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=257">Encyclopedia of Combinatorial Structures 257</a>.

%H Milan Janjic, <a href="https://old.pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">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., Vol. 131, No. 1 (2002), pp. 65-75.

%H 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 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 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 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 The MacTutor History of Mathematics archive, <a href="http://www-history.mcs.st-and.ac.uk/Mathematicians/Narayana.html">Narayana Pandit</a>.

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

%H <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 G.f.: x^7/(1-x)^8.

%F a(n) = (n^7 - 21*n^6 + 175*n^5 - 735*n^4 + 1624*n^3 - 1764*n^2 + 720*n)/5040.

%F a(n) = -A110555(n+1,7). - _Reinhard Zumkeller_, Jul 27 2005

%F 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 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 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 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 a(n) = 3*C(n+1,7) = 3*A000580(n+1). - _Serhat Bulut_, Mar 13 2015

%F From _Wolfdieter Lang_, Mar 21 2015: (Start)

%F a(n) = A104712(n, 7), n >= 7.

%F a(n+6) = sum(A000579(j+5), j = 1..n), n >= 1. See the Mar 20 2015 comment above. (End)

%F Sum_{k >= 7} 1/a(k) = 7/6. - _Tom Edgar_, Sep 10 2015

%F 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 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 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 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:=1/(z-1)**8; # _Simon Plouffe_ in his 1992 dissertation, offset 0.

%p seq(binomial(n+7,7)*1^n,n=0..30); # _Zerinvary Lajos_, Jun 23 2008

%p 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 Table[n(n + 1)(n + 2)(n + 3)(n + 4)(n + 5)(n + 6)/7!, {n, 1, 100}] (* _Artur Jasinski_, Dec 02 2007 *)

%t 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 CoefficientList[Series[1 / (1-x)^8, {x, 0, 33}], x] (* _Vincenzo Librandi_, Mar 21 2015 *)

%o (Magma) [Binomial(n,7): n in [7..40]]; // _Vincenzo Librandi_, Mar 21 2015

%o (PARI) a(n)=binomial(n,7) \\ _Charles R Greathouse IV_, Sep 24 2015

%Y Cf. A053136, A053129, A000579, A000581, A000582, A000217, A000292, A000332, A000389, A104712, A001787, A242091.

%K nonn,easy

%O 7,2

%A _N. J. A. Sloane_

%E More terms from Larry Reeves (larryr(AT)acm.org), Mar 17 2000

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