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A000579 Figurate numbers or binomial coefficients C(n,6).
(Formerly M4390 N1847)
59
0, 0, 0, 0, 0, 0, 1, 7, 28, 84, 210, 462, 924, 1716, 3003, 5005, 8008, 12376, 18564, 27132, 38760, 54264, 74613, 100947, 134596, 177100, 230230, 296010, 376740, 475020, 593775, 736281, 906192, 1107568, 1344904, 1623160, 1947792, 2324784, 2760681, 3262623 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

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

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.

Philippe A. J. G. Chevalier, On the discrete geometry of physical quantities, Preprint, 2012. - From N. J. A. Sloane, Aug 28 2012

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

LINKS

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 to sequences with linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

FORMULA

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

EXAMPLE

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 + ...

MAPLE

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

MATHEMATICA

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 *)

PROG

(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

CROSSREFS

Cf. A053135, A053128, A000580, A000581, A000582, A000217, A000292, A000332, A000389.

Sequence in context: A049018 A008489 A023032 * A049017 A019501 A229887

Adjacent sequences:  A000576 A000577 A000578 * A000580 A000581 A000582

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

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Last modified December 21 15:47 EST 2014. Contains 252324 sequences.