A001288 a(n) = binomial(n,11).

1, 12, 78, 364, 1365, 4368, 12376, 31824, 75582, 167960, 352716, 705432, 1352078, 2496144, 4457400, 7726160, 13037895, 21474180, 34597290, 54627300, 84672315, 129024480, 193536720, 286097760, 417225900, 600805296, 854992152, 1203322288, 1676056044

Offset: 11

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Product of 11 consecutive numbers divided by 11!. - Artur Jasinski, Dec 02 2007

In this sequence there are no primes. - Artur Jasinski, Dec 02 2007

With a different offset, number of n-permutations (n>=11) of 2 objects: u,v, with repetition allowed, containing exactly (11) u's. Example: n=11, a(0)=1 because we have uuuuuuuuuuu n=12, a(1)=12 because we have uuuuuuuuuuuv, uuuuuuuuuuvu, uuuuuuuuuvuu, uuuuuuuuvuuu, uuuuuuuvuuuu, uuuuuuvuuuuu, uuuuuvuuuuuu, uuuuvuuuuuuu, uuuvuuuuuuuu, uuvuuuuuuuuu uvuuuuuuuuuu, vuuuuuuuuuuu. - Zerinvary Lajos, Aug 06 2008

Does not satisfy Benford's law (because n^11 does not, see Ross, 2012). - N. J. A. Sloane, Feb 09 2017

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=11..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].
• Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
• A. S. Chinchon, Mixing Benford, GoogleVis And On-Line Encyclopedia of Integer Sequences, 2014. Note: as of Feb 09 2017, the results in this page appear to be incorrect - N. J. A. Sloane, Feb 09 2017.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 261
• Milan Janjic, Two Enumerative Functions.
• Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 15.
• Ângela Mestre, José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
• Kenneth A. Ross, First Digits of Squares and Cubes, Math. Mag. 85 (2012) 36-42.
• Index entries for sequences related to Benford's law
• Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Crossrefs

Cf. A110555, A001787, A242091.

Programs

Maple

seq(binomial(n,11),n=0..30); # Zerinvary Lajos, Aug 06 2008, R. J. Mathar, Jul 07 2009

Mathematica

Table[n(n+1)(n+2)(n+3)(n+4)(n+5)(n+6)(n+7)(n+8)(n+9)(n+10)/11!,{n,1,100}] (* Artur Jasinski, Dec 02 2007 *)
Binomial[Range[11,50],11] (* Harvey P. Dale, Oct 02 2012 *)

PARI

for(n=11, 50, print1(binomial(n,11), ", ")) \\ G. C. Greubel, Aug 31 2017

Formula

a(n) = -A110555(n+1,11). - Reinhard Zumkeller, Jul 27 2005

a(n+10) = n(n+1)(n+2)(n+3)(n+4)(n+5)(n+6)(n+7)(n+8)(n+9)(n+10)/11!. - Artur Jasinski, Dec 02 2007; R. J. Mathar, Jul 07 2009

G.f.: x^11/(1-x)^12. a(n) = binomial(n,11). - Zerinvary Lajos, Aug 06 2008; R. J. Mathar, Jul 07 2009

From Amiram Eldar, Dec 10 2020: (Start)

Sum_{n>=11} 1/a(n) = 11/10.

Sum_{n>=11} (-1)^(n+1)/a(n) = A001787(11)*log(2) - A242091(11)/10! = 11264*log(2) - 491821/63 = 0.9273021446... (End)

Extensions

Some formulas for other offsets corrected by R. J. Mathar, Jul 07 2009