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A000579
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Figurate numbers or binomial coefficients C(n,6).
(Formerly M4390 N1847)
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50
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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; internal format)
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OFFSET
| 0,8
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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 (jvospost3(AT)gmail.com), Nov 28 2004
a(n) = A110555(n+1,6). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 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 (sfalcon(AT)dma.ulpgc.es), Feb 12 2007
Only prime in this sequence is 7 - Artur Jasinski (grafix(AT)csl.pl), Dec 02 2007
6-dimensional triangular numbers, sixth partial sums of binomial transform of [1,0,0,0,...]. [From 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 (shevelev(AT)bgu.ac.il), 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
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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.
Leo Moser, Mathematics Magazine, 26 (March, 1953), p. 226.
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
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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].
P. 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.
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
J. V. Post, Table of Polytope Numbers, Sorted, Through 1,000,000.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Index entries for sequences related to linear recurrences with constant coefficients
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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 (ralf(AT)ark.in-berlin.de), 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 (sfalcon(AT)dma.ulpgc.es), Feb 12 2007
a(n)=n(n-1)(n-2)(n-3)(n-4)(n-5)/720 - Artur Jasinski (grafix(AT)csl.pl), Dec 02 2007, R. J. Mathar, Jul 07 2009
Equals binomial transform of [1, 6, 15, 20, 15, 6, 1, 0, 0, 0,...]. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 02 2008]
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EXAMPLE
| a(4) = 84 = (1, 3, 3, 1) dot (1, 6, 15, 20) = (1 + 18 + 45 + 20). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 02 2008]
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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 (zerinvarylajos(AT)yahoo.com), Mar 13 2007
A000579:=-1/(z-1)**7; [S. Plouffe in his 1992 dissertation, referring to offset 0.]
seq(binomial(n, 6), n=0..33); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 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); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 05 2009]
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MATHEMATICA
| Table[Binomial[n, 6], {n, 6, 50}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 02 2006
Table[n(n - 1)(n - 2)(n - 3)(n - 4)(n - 5)/720, {n, 0, 100}] - Artur Jasinski (grafix(AT)csl.pl), Dec 02 2007
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CROSSREFS
| Cf. A053135, A053128, A000580, A000581, A000582.
Cf. A000217, A000292, A000332, A000389.
Sequence in context: A049018 A008489 A023032 * A049017 A019501 A145456
Adjacent sequences: A000576 A000577 A000578 * A000580 A000581 A000582
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Larry Reeves (larryr(AT)acm.org), Mar 17 2000
More terms from Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 02 2006
Some formulas that referred to other offsets corrected by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 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 (mathar(AT)strw.leidenuniv.nl), Aug 03 2010
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