Search: id:A000579
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%I A000579 M4390 N1847
%S A000579 1,7,28,84,210,462,924,1716,3003,5005,8008,12376,18564,27132,38760,
%T A000579 54264,74613,100947,134596,177100,230230,296010,376740,475020,593775,
%U A000579 736281,906192,1107568,1344904,1623160,1947792,2324784,2760681,3262623
%N A000579 Figurate numbers or binomial coefficients C(n,6).
%C A000579 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.
%C A000579 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
%C A000579 a(n) = A110555(n+1,6). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Jul 27 2005
%C A000579 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
%C A000579 Product of six consecutive numbers divided by 6! - Artur Jasinski (grafix(AT)csl.pl), 
               Dec 02 2007
%C A000579 Only prime in this sequence is 7 - Artur Jasinski (grafix(AT)csl.pl), 
               Dec 02 2007
%C A000579 With a different offset, number of n-permutations (n>=6) of 2 objects: 
               u,v, with repetition allowed, containing exactly six (6) u's. Example: 
               a(1)=7 because we have uuuuuuv, uuuuuvu, uuuuvuu, uuuvuuu, uuvuuuu, 
               uvuuuuu and vuuuuuu. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jun 16 2008
%C A000579 6-dimensional triangular numbers, sixth partial sums of binomial transform 
               of [1,0,0,0,...]. a(n+6)=sum{i=0,n,C(n+6,i+6)*b(i)}, where b(i)=[1,
               0,0,0,...], a(n+6)=C(n+6,6). [From Borislav St. Borisov (b.st.borisov(AT)abv.bg), 
               Mar 05 2009, R. J. Mathar, Jul 07 2009]
%D A000579 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000579 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000579 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.
%D A000579 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 A000579 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 A000579 Leo Moser, Mathematics Magazine, 26 (March, 1953), p. 226.
%D A000579 J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society 
               Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
%D A000579 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, 
               p. 196.
%D A000579 Charles W. Trigg: Mathematical Quickies. New York: Dover Publications, 
               Inc., 1985, p. 11, #32
%H A000579 T. D. Noe, <a href="b000579.txt">Table of n, a(n) for n=6..1000</a>
%H A000579 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A000579 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative 
               Functions</a>
%H A000579 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A000579 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A000579 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A000579 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Sequences realized by oligomorphic permutation groups</a>, J. Integ. 
               Seqs. Vol. 3 (2000), #00.1.5.
%H A000579 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=256">
               Encyclopedia of Combinatorial Structures 256</a>
%H A000579 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Composition.html">Link to a section of The World of Mathematics.</
               a>
%H A000579 H. K. Kim, <a href="http://com2mac.postech.ac.kr/papers/2001/01-22.pdf">
               On Regular Polytope Numbers</a>, Journal: Proc. Amer.Math. Soc. 131 
               (2003), 65-75, as PDF file.
%H A000579 J. V. Post, <a href="http://www.magicdragon.com/poly.html">Table of Polytope 
               Numbers, Sorted, Through 1,000,000</a>.
%F A000579 G.f.: x^6/(1-x)^7.
%F A000579 (x^6-15*x^5+85*x^4-225*x^3+274*x^2-120*x)/720
%F A000579 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
%F A000579 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
%F A000579 a(n)=numbperm (n,6)/720, n>=6 . - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Apr 26 2007
%F A000579 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
%F A000579 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]
%e A000579 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]
%p A000579 A000579 := n->binomial(n,6);
%p A000579 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
%p A000579 seq(numbperm (n,6)/720, n=6..39); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Apr 26 2007
%p A000579 A000579:=-1/(z-1)**7; [S. Plouffe in his 1992 dissertation, referring 
               to offset 0.]
%p A000579 seq(binomial(n+6,6)*1^n,n=0..33); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jun 16 2008
%p A000579 restart: 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]
%t A000579 Table[Binomial[n, 6], {n, 6, 50}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), 
               Apr 02 2006
%t A000579 Table[n(n + 1)(n + 2)(n + 3)(n + 4)(n + 5)/720, {n, 1, 100}] - Artur 
               Jasinski (grafix(AT)csl.pl), Dec 02 2007
%Y A000579 Cf. A053135, A053128, A000580, A000581, A000582.
%Y A000579 Cf. A000217, A000292, A000332, A000389.
%Y A000579 Sequence in context: A049018 A008489 A023032 this_sequence A049017 A019501 
               A145456
%Y A000579 Adjacent sequences: A000576 A000577 A000578 this_sequence A000580 A000581 
               A000582
%K A000579 nonn,easy,nice
%O A000579 6,2
%A A000579 N. J. A. Sloane (njas(AT)research.att.com).
%E A000579 More terms from Larry Reeves (larryr(AT)acm.org), Mar 17 2000
%E A000579 More terms from Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), 
               Apr 02 2006
%E A000579 Some formulas that referred to other offsets corrected by R. J. Mathar 
               (mathar(AT)strw.leidenuniv.nl), Jul 07 2009

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