Search: id:A000580
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%I A000580 M4517 N1911
%S A000580 1,8,36,120,330,792,1716,3432,6435,11440,19448,31824,50388,77520,
%T A000580 116280,170544,245157,346104,480700,657800,888030,1184040,1560780,
%U A000580 2035800,2629575,3365856,4272048,5379616,6724520,8347680,10295472
%N A000580 Binomial coefficients C(n,7).
%C A000580 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 (jvospost3(AT)gmail.com), Nov 28 2004
%C A000580 a(n) = -A110555(n+1,7). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Jul 27 2005
%C A000580 a(n) is the number of terms in the expansion of ( sum_{i=1}^8 a_i)^n 
               - Sergio Falcon (sfalcon(AT)dma.ulpgc.es), Feb 12 2007
%C A000580 Product of seven consecutive numbers divided by 7! - Artur Jasinski (grafix(AT)csl.pl), 
               Dec 02 2007
%C A000580 In this sequence there are no primes - Artur Jasinski (grafix(AT)csl.pl), 
               Dec 02 2007
%C A000580 With a different offset, number of n-permutations (n>=7) of 2 objects: 
               u,v with repetition allowed, containing exactly seven (7) u's. Example: 
               a(1)=8 because we have uuuuuuuv, uuuuuuvu, uuuuuvuu, uuuuvuuu, uuuvuuuu, 
               uuvuuuuu, uvuuuuuu and vuuuuuuu. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jun 23 2008
%D A000580 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000580 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000580 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 A000580 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, 
               p. 196.
%D A000580 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 A000580 J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society 
               Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
%H A000580 T. D. Noe, <a href="b000580.txt">Table of n, a(n) for n=7..1000</a>
%H A000580 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 A000580 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 A000580 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=257">
               Encyclopedia of Combinatorial Structures 257</a>
%H A000580 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative 
               Functions</a>
%H A000580 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 A000580 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 A000580 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 A000580 J. V. Post, <a href="http://www.magicdragon.com/poly.html">Table of Polytope 
               Numbers, Sorted, Through 1,000,000</a>.
%H A000580 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 A000580 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%F A000580 G.f.: x^7/(1-x)^8.
%F A000580 (x^7-21*x^6+175*x^5-735*x^4+1624*x^3-1764*x^2+720*x)/5040.
%F A000580 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 (sfalcon(AT)dma.ulpgc.es), Feb 
               12 2007
%F A000580 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. W. Lang, 
               Apr 04 2007.
%F A000580 a(n)=n(n-1)(n-2)(n-3)(n-4)(n-5)(n-6)/7! - Artur Jasinski (grafix(AT)csl.pl), 
               Dec 02 2007, R. J. Mathar, Jul 07 2009
%p A000580 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 (zerinvarylajos(AT)yahoo.com), 
               Mar 13 2007
%p A000580 A000580:=1/(z-1)**8; [S. Plouffe in his 1992 dissertation, offset 0.]
%p A000580 seq(binomial(n+7,7)*1^n,n=0..30); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jun 23 2008
%p A000580 restart: 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);# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Apr 05 2009]
%t A000580 Table[n(n + 1)(n + 2)(n + 3)(n + 4)(n + 5)(n + 6)/7!, {n, 1, 100}] - 
               Artur Jasinski (grafix(AT)csl.pl), Dec 02 2007
%Y A000580 Cf. A053136, A053129, A000579, A000581, A000582.
%Y A000580 Cf. A000217, A000292, A000332, A000389, A000579.
%Y A000580 Sequence in context: A008500 A008490 A023033 this_sequence A145457 A145136 
               A144901
%Y A000580 Adjacent sequences: A000577 A000578 A000579 this_sequence A000581 A000582 
               A000583
%K A000580 nonn,easy
%O A000580 7,2
%A A000580 N. J. A. Sloane (njas(AT)research.att.com).
%E A000580 More terms from Larry Reeves (larryr(AT)acm.org), Mar 17 2000
%E A000580 Some formulas that referred to other offsets corrected by R. J. Mathar 
               (mathar(AT)strw.leidenuniv.nl), Jul 07 2009

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