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A000580
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Binomial coefficients C(n,7).
(Formerly M4517 N1911)
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37
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1, 8, 36, 120, 330, 792, 1716, 3432, 6435, 11440, 19448, 31824, 50388, 77520, 116280, 170544, 245157, 346104, 480700, 657800, 888030, 1184040, 1560780, 2035800, 2629575, 3365856, 4272048, 5379616, 6724520, 8347680, 10295472
(list; graph; refs; listen; history; internal format)
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OFFSET
| 7,2
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COMMENTS
| 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
a(n) = -A110555(n+1,7). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jul 27 2005
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
Product of seven consecutive numbers divided by 7! - Artur Jasinski (grafix(AT)csl.pl), Dec 02 2007
In this sequence there are no primes - Artur Jasinski (grafix(AT)csl.pl), Dec 02 2007
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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.
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).
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LINKS
| T. D. Noe, Table of n, a(n) for n=7..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.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 257
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^7/(1-x)^8.
(x^7-21*x^6+175*x^5-735*x^4+1624*x^3-1764*x^2+720*x)/5040.
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
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.
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
a(7)=1, a(8)=8, a(9)=36, a(10)=120, a(11)=330, a(12)=792, a(13)=1716, a(14)=3432, a(n)=8*a(n-1)-28*a(n-2)+56*a(n-3)-70*a(n-4)+56*a(n-5)- 28*a(n-6)+8*a(n-7)-a(n-8) [From Harvey P. Dale, Nov 28 2011]
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MAPLE
| 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
A000580:=1/(z-1)**8; [S. Plouffe in his 1992 dissertation, offset 0.]
seq(binomial(n+7, 7)*1^n, n=0..30); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 23 2008
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]
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MATHEMATICA
| 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
Binomial[Range[7, 40], 7] (* or *) LinearRecurrence[ {8, -28, 56, -70, 56, -28, 8, -1}, {1, 8, 36, 120, 330, 792, 1716, 3432}, 40] (* From Harvey P. Dale, Nov 28 2011 *)
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CROSSREFS
| Cf. A053136, A053129, A000579, A000581, A000582.
Cf. A000217, A000292, A000332, A000389, A000579.
Sequence in context: A008500 A008490 A023033 * A145457 A145136 A144901
Adjacent sequences: A000577 A000578 A000579 * A000581 A000582 A000583
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KEYWORD
| nonn,easy
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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
Some formulas that referred to other offsets corrected by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 07 2009
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