0,8

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

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

a(n) = A110555(n+1,6). - Reinhard Zumkeller, Jul 27 2005

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

Only prime in this sequence is 7. - Artur Jasinski, Dec 02 2007

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

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

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

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

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

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.

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

P. A. J. G. Chevalier, On a Mathematical Method for Discovering Relations Between Physical Quantities: a Photonics Case Study, Slides from a talk presented at ICOL2014, 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

P. A. J. G. Chevalier, A “table of Mendeleev” for physical quantities?, Slides from a talk, May 14 2014, Leuven, Belgium, http://www.researchgate.net/profile/Philippe_Chevalier2/publication/262067273_A_table_of_Mendeleev_for_physical_quantities/links/0c9605368f6d191478000000.pdf

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.

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).

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

T. D. Noe, Table of n, a(n) for n = 0..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].

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

Michael B. Green, Stephen D. Miller, and Pierre Vanhove, Small representations, string instantons, and Fourier modes of Eisenstein series

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 256

Milan Janjic, Two Enumerative Functions

H. K. Kim, On Regular Polytope Numbers, Journal: Proc. Amer.Math. Soc. 131 (2003), 65-75, as PDF file.

Leo Moser, Quicky 87, Mathematics Magazine, 26 (March 1953), p. 226.

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.

Jonathan Vos Post, Table of Polytope Numbers, Sorted, Through 1,000,000.

Hermann Stamm-Wilbrandt, Sum of Pascal's triangle reciprocals

Eric Weisstein's World of Mathematics, Composition

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Serhat Bulut, Oktay Erkan Temizkan, Subset Sum Problem

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

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

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

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

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

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

Equals binomial transform of [1, 6, 15, 20, 15, 6, 1, 0, 0, 0,...]. - Gary W. Adamson, Aug 02 2008

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

sum_{n >= 6} 1/a(n) = 6/5. - Hermann Stamm-Wilbrandt, Jul 13 2014

sum(n >= 6, (-1)^(n + 1)/a(n)) = 192*log(2) - 661/5  = 0.8842586675... Also see A242023. - Richard R. Forberg, Aug 11 2014

a(n) = a(5-n) for all n in Z. - Michael Somos, Oct 07 2014

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

a(n) = 3*C(n+1,6) = 3* A000579(n+1). - Serhat Bulut, Oktay Erkan Temizkan, Mar 13 2015

a(4) = 84 = (1, 3, 3, 1) dot (1, 6, 15, 20) = (1 + 18 + 45 + 20). - Gary W. Adamson, Aug 02 2008

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

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

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

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

A000579:=-1/(z-1)**7; # Simon Plouffe in his 1992 dissertation, referring to offset 0.

seq(binomial(n, 6), n=0..33); # Zerinvary Lajos, Jun 16 2008

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

Table[Binomial[n, 6], {n, 6, 50}] (* Stefan Steinerberger, Apr 02 2006 *)

Table[n(n - 1)(n - 2)(n - 3)(n - 4)(n - 5)/720, {n, 0, 100}] (* Artur Jasinski, Dec 02 2007 *)

LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 0, 0, 0, 0, 0, 1}, 50] (* Harvey P. Dale, Dec 30 2012 *)

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

(PARI) a(n)=binomial(n, 6) \\ Charles R Greathouse IV, Nov 20 2012

(MAGMA) [Binomial(n, 6) : n in [0..50]]; // Wesley Ivan Hurt, Jul 13 2014

Cf. A053135, A053128, A000580, A000581, A000582, A000217, A000292, A000332, A000389, A104712 (fifth column, k=6).

Sequence in context: A049018 A008489 A023032 * A049017 A019501 A229887

Adjacent sequences:  A000576 A000577 A000578 * A000580 A000581 A000582

nonn,easy,nice,changed

N. J. A. Sloane

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

I changed the offset to 0. This will require some further adjustments to the formulas. - N. J. A. Sloane, Aug 01 2010

Inserted Shevelev comment, further adaptations to offset - R. J. Mathar, Aug 03 2010

approved