OFFSET
10,2
COMMENTS
Coordination sequence for 10-dimensional cyclotomic lattice Z[zeta_11].
Product of 10 consecutive numbers divided by 10!. - Artur Jasinski, Dec 02 2007
In this sequence only 11 is prime. - Artur Jasinski, Dec 02 2007
With a different offset, number of n-permutations (n>=10) of 2 objects: u,v, with repetition allowed, containing exactly 10 u's. Example: a(1)=11 because we have uuuuuuuuuuv, uuuuuuuuuvu, uuuuuuuuvuu, uuuuuuuvuuu, uuuuuuvuuuu, uuuuuvuuuuu, uuuuvuuuuuu, uuuvuuuuuuu, uuvuuuuuuuu, uvuuuuuuuuu and vuuuuuuuuuu. - Zerinvary Lajos, Aug 03 2008
a(9+k) is the number of times that each digit appears repeated inside a list made with all the possible base 10 numbers of k digits such that their digits are read in ascending order from left to right. - R. J. Cano Jul 20 2014
a(n) = fallfac(n,10)/10! = binomial(n, 10) is also the number of independent components of an antisymmetric tensor of rank 10 and dimension n >= 10 (for n=1..9 this becomes 0). Here fallfac is the falling factorial. - Wolfdieter Lang, Dec 10 2015
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.
Albert 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).
LINKS
T. D. Noe, Table of n, a(n) for n = 10..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].
Matthias Beck and Serkan Hosten, Cyclotomic polytopes and growth series of cyclotomic lattices, arXiv:math/0508136 [math.CO], 2005-2006.
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. II. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 9.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 260.
Milan Janjic, Two Enumerative Functions.
Rajesh Kumar Mohapatra and Tzung-Pei Hong, On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences, Mathematics (2022) Vol. 10, No. 7, 1161.
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).
FORMULA
a(n) = A110555(n+1,10). - Reinhard Zumkeller, Jul 27 2005
a(n+9) = n(n+1)(n+2)(n+3)(n+4)(n+5)(n+6)(n+7)(n+8)(n+9)/10!. - Artur Jasinski, Dec 02 2007; R. J. Mathar, Jul 07 2009
G.f.: x^10/(1-x)^11. - Zerinvary Lajos, Aug 06 2008; R. J. Mathar, Jul 07 2009
Sum_{k>=10} 1/a(k) = 10/9. - Tom Edgar, Sep 10 2015
Sum_{n>=10} (-1)^n/a(n) = A001787(10)*log(2) - A242091(10)/9! = 5120*log(2) - 447047/126 = 0.9215009748... - Amiram Eldar, Dec 10 2020
MAPLE
seq(binomial(n, 10), n=10..31); # Zerinvary Lajos, Aug 06 2008
MATHEMATICA
Table[n (n + 1) (n + 2) (n + 3) (n + 4) (n + 5) (n + 6) (n + 7) (n + 8) (n + 9)/10!, {n, 1, 100}] (* Artur Jasinski, Dec 02 2007 *)
Table[Binomial[n, 10], {n, 10, 20}] (* Zerinvary Lajos, Jan 31 2010 *)
PROG
(Magma) [Binomial(n, 10): n in [10..40]]; // Vincenzo Librandi, Sep 11 2015
(PARI) a(n)=binomial(n, 10) \\ Charles R Greathouse IV, Sep 24 2015
(Python)
A001287_list, m = [], [1]*11
for _ in range(10**2):
A001287_list.append(m[-1])
for i in range(10):
m[i+1] += m[i] # Chai Wah Wu, Jan 24 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Formulas valid for different offsets rewritten by R. J. Mathar, Jul 07 2009
Extended by Ray Chandler, Oct 25 2011
STATUS
approved