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A000580 Binomial coefficients C(n,7).
(Formerly M4517 N1911)
55
1, 8, 36, 120, 330, 792, 1716, 3432, 6435, 11440, 19448, 31824, 50388, 77520, 116280, 170544, 245157, 346104, 480700, 657800, 888030, 1184040, 1560780, 2035800, 2629575, 3365856, 4272048, 5379616, 6724520, 8347680, 10295472 (list; graph; refs; listen; history; text; internal format)
OFFSET

7,2

COMMENTS

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

a(n) = -A110555(n+1,7). - Reinhard Zumkeller, Jul 27 2005

a(n) is the number of terms in the expansion of ( sum_{i=1}^8 a_i)^n. - Sergio Falcon, Feb 12 2007

Product of seven consecutive numbers divided by 7!. - Artur Jasinski, Dec 02 2007

In this sequence there are no primes. - Artur Jasinski, Dec 02 2007

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

Partial sums of A000579. In general, the iterated sums S(m, n) = sum(S(m-1, j), j=1..n) 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(risefac(k, m)/m!, k = 1..n), with the rising factorials risefac(x, m):=  product(x+j, j=0..m-1), 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

Sum_{k >= 7} 1/a(k) = 7/6. - Tom Edgar, Sep 10 2015

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

Number of compositions (ordered partitions) of n+1 into exactly 8 parts. Juergen Will, Jan 02 2016

Number of weak compositions (ordered weak partitions) of n-7 into exactly 8 parts. - Juergen Will, Jan 02 2016

REFERENCES

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.

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 196.

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.

S. Gottwald et al. (eds.), Lexikon bedeutender Mathematiker (in German), Bibliographisches Institut Leipzig, 1990.

J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n = 7..1000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Serhat Bulut and Oktay Erkan Temizkan, Subset Sum Problem, 2015

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

Philippe A. J. G. Chevalier, On the discrete geometry of physical quantities, Preprint, 2012.

P. A. J. G. Chevalier, A "table of Mendeleev" for physical quantities?, Slides from a talk, May 14 2014, Leuven, Belgium.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 257

Milan Janjic, Two Enumerative Functions

Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2002), 65-75.

P. A. MacMahon, Memoir on the Theory of the Compositions of Numbers, Phil. Trans. Royal Soc. London A, 184 (1893), 835-901. - Juergen Will, Jan 02 2016

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

J. V. Post, Table of Polytope Numbers, Sorted, Through 1,000,000.

The MacTutor History of Mathematics archive, Narayana Pandit.

Eric Weisstein's World of Mathematics, Composition.

Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

FORMULA

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

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

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

a(n+4) = (1/3!)*diff(S(n,x),x$3)|_{x=2}, n>=3. One sixth of third derivative of Chebyshev S-polynomials evaluated at x=2. See A049310. - Wolfdieter Lang, Apr 04 2007

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

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

a(n) = 3*C(n+1,7) = 3*A000580(n+1). - Serhat Bulut, Mar 13 2015

From Wolfdieter Lang, Mar 21 2015: (Start)

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

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

EXAMPLE

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}.

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

MAPLE

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

A000580:=1/(z-1)**8; # Simon Plouffe in his 1992 dissertation, offset 0.

seq(binomial(n+7, 7)*1^n, n=0..30); # Zerinvary Lajos, Jun 23 2008

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

MATHEMATICA

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

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 *)

CoefficientList[Series[1 / (1-x)^8, {x, 0, 33}], x] (* Vincenzo Librandi, Mar 21 2015 *)

PROG

(MAGMA) [Binomial(n, 7): n in [7..40]]; // Vincenzo Librandi, Mar 21 2015

(PARI) a(n)=binomial(n, 7) \\ Charles R Greathouse IV, Sep 24 2015

CROSSREFS

Cf. A053136, A053129, A000579, A000581, A000582, A000217, A000292, A000332, A000389, A104712.

Sequence in context: A008500 A008490 A023033 * A229888 A243742 A145457

Adjacent sequences:  A000577 A000578 A000579 * A000581 A000582 A000583

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

Some formulas that referred to other offsets corrected by R. J. Mathar, Jul 07 2009

STATUS

approved

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Last modified June 24 16:30 EDT 2016. Contains 274185 sequences.