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A001287
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a(n) = binomial coefficient C(n,10).
(Formerly M4794 N2046)
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20
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1, 11, 66, 286, 1001, 3003, 8008, 19448, 43758, 92378, 184756, 352716, 646646, 1144066, 1961256, 3268760, 5311735, 8436285, 13123110, 20030010, 30045015, 44352165, 64512240, 92561040, 131128140, 183579396, 254186856, 348330136, 472733756, 635745396
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OFFSET
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10,2
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COMMENTS
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Coordination sequence for 10-dimensional cyclotomic lattice Z[zeta_11].
Product of 10 consecutive numbers divided by 10!. - 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
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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.
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).
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LINKS
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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].
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).
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FORMULA
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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
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MAPLE
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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) (n + 9)/10!, {n, 1, 100}] (* Artur Jasinski, Dec 02 2007 *)
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PROG
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(Python)
for _ in range(10**2):
for i in range(10):
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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Formulas valid for different offsets rewritten by R. J. Mathar, Jul 07 2009
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STATUS
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approved
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