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A000582
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Binomial coefficients C(n,9).
(Formerly M4712 N2013)
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25
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1, 10, 55, 220, 715, 2002, 5005, 11440, 24310, 48620, 92378, 167960, 293930, 497420, 817190, 1307504, 2042975, 3124550, 4686825, 6906900, 10015005, 14307150, 20160075, 28048800, 38567100, 52451256, 70607460, 94143280, 124403620, 163011640, 211915132
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OFFSET
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9,2
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COMMENTS
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Figurate numbers based on 9-dimensional regular simplex. - Jonathan Vos Post, Nov 28 2004
a(n) = -A110555(n+1, 9). - Reinhard Zumkeller, Jul 27 2005
Product of 9 consecutive numbers divided by 9!. - Artur Jasinski, Dec 02 2007
In this sequence there are no primes. - Artur Jasinski, Dec 02 2007
a(9+n) counts the number of words with n letters over the alphabet {0,1,..,9} such that these letters are read from left to right in weakly increasing (non-decreasing) order. - R. J. Cano, Jul 20 2014
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REFERENCES
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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
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T. D. Noe, Table of n, a(n) for n = 9..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 259
Milan Janjic, Two Enumerative Functions
H. K. Kim, On Regular Polytope Numbers, Journal: Proc. Amer. Math. Soc. 131 (2003), 65-75, as PDF file.
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.
J. V. Post, Table of Polytope Numbers, Sorted, Through 1,000,000.
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FORMULA
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G.f.: x^9/(1-x)^10.
a(n+8) = n(n+1)(n+2)(n+3)(n+4)(n+5)(n+6)(n+7)(n+8)/9!. - Artur Jasinski, Dec 02 2007, R. J. Mathar, Jul 07 2009
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MAPLE
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A000582 := n->binomial(n, 9): seq(A000582(n), n=9..40);
A000582:=1/(z-1)**10; # Simon Plouffe in his 1992 dissertation (offset 0)
seq(binomial(n, 9), n=0..29); # Zerinvary Lajos, Jun 23 2008, R. J. Mathar, Jul 07 2009
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MATHEMATICA
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Table[n(n+1)(n+2)(n+3)(n+4)(n+5)(n+6)(n+7)(n+8)/9!, {n, 100}] (* Artur Jasinski, Dec 02 2007 *)
Table[Binomial[n, 9], {n, 9, 50}] (* Wesley Ivan Hurt, Jul 20 2014 *)
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PROG
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(MAGMA) [Binomial(n, 9) : n in [9..50]]; // Wesley Ivan Hurt, Jul 20 2014
(PARI) a(n)=binomial(n, 9) \\ Charles R Greathouse IV, Jul 21 2014
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CROSSREFS
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Cf. A053138, A053131, A000581, A035927.
Sequence in context: A008492 A023035 A128936 * A229890 A243744 A145459
Adjacent sequences: A000579 A000580 A000581 * A000583 A000584 A000585
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KEYWORD
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easy,nonn
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from Winston C. Yang (winston(AT)cs.wisc.edu), Aug 23 2000
Formulas referring to other offsets rewritten by R. J. Mathar, Jul 07 2009
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STATUS
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approved
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