%I M4850 N2073 #67 Dec 15 2023 15:16:56

%S 1,12,78,364,1365,4368,12376,31824,75582,167960,352716,705432,1352078,

%T 2496144,4457400,7726160,13037895,21474180,34597290,54627300,84672315,

%U 129024480,193536720,286097760,417225900,600805296,854992152,1203322288,1676056044

%N a(n) = binomial(n,11).

%C Product of 11 consecutive numbers divided by 11!. - _Artur Jasinski_, Dec 02 2007

%C In this sequence there are no primes. - _Artur Jasinski_, Dec 02 2007

%C 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

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

%D 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.

%D Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 196.

%D 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.

%D J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A001288/b001288.txt">Table of n, a(n) for n=11..1000</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H Peter J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

%H A. S. Chinchon, <a href="https://fronkonstin.com/2014/12/23/mixing-bendford-googlevis-and-on-line-encyclopedia-of-integer-sequences/">Mixing Benford, GoogleVis And On-Line Encyclopedia of Integer Sequences</a>, 2014. Note: as of Feb 09 2017, the results in this page appear to be incorrect - N. J. A. Sloane, Feb 09 2017.

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=261">Encyclopedia of Combinatorial Structures 261</a>

%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>.

%H Ângela Mestre, José Agapito, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Mestre/mestre2.html">Square Matrices Generated by Sequences of Riordan Arrays</a>, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.

%H Kenneth A. Ross, <a href="http://www.jstor.org/stable/10.4169/math.mag.85.1.036">First Digits of Squares and Cubes</a>, Math. Mag. 85 (2012) 36-42.

%H <a href="/index/Be#Benford">Index entries for sequences related to Benford's law</a>

%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (12, -66, 220, -495, 792, -924, 792, -495, 220, -66, 12, -1).

%F a(n) = -A110555(n+1,11). - _Reinhard Zumkeller_, Jul 27 2005

%F 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

%F G.f.: x^11/(1-x)^12. a(n) = binomial(n,11). - _Zerinvary Lajos_, Aug 06 2008; _R. J. Mathar_, Jul 07 2009

%F From _Amiram Eldar_, Dec 10 2020: (Start)

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

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

%p seq(binomial(n,11),n=0..30); # _Zerinvary Lajos_, Aug 06 2008, _R. J. Mathar_, Jul 07 2009

%t 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 *)

%t Binomial[Range[11,50],11] (* _Harvey P. Dale_, Oct 02 2012 *)

%o (PARI) for(n=11, 50, print1(binomial(n,11), ", ")) \\ _G. C. Greubel_, Aug 31 2017

%Y Cf. A110555, A001787, A242091.

%K nonn

%O 11,2

%A _N. J. A. Sloane_

%E Some formulas for other offsets corrected by _R. J. Mathar_, Jul 07 2009