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%I M4522 N1916 #112 Sep 08 2022 08:44:29
%S 1,8,40,160,560,1792,5376,15360,42240,112640,292864,745472,1863680,
%T 4587520,11141120,26738688,63504384,149422080,348651520,807403520,
%U 1857028096,4244635648,9646899200,21810380800,49073356800,109924319232
%N a(n) = binomial(n,3)*2^(n-3).
%C Number of 3-dimensional cubes in n-dimensional hypercube. - _Henry Bottomley_, Apr 14 2000
%C With three leading zeros, this is the second binomial transform of (0,0,0,1,0,0,0,0,...). - _Paul Barry_, Mar 07 2003
%C With 3 leading zeros, binomial transform of C(n,3). - _Paul Barry_, Apr 10 2003
%C Let M=[1,0,i;0,1,0;i,0,1], i=sqrt(-1). Then 1/det(I-xM)=1/(1-2x)^4. - _Paul Barry_, Apr 27 2005
%C If X_1,X_2,...,X_n is a partition of a 2n-set X into 2-blocks then, for n>2, a(n+1) is equal to the number of (n+3)-subsets of X intersecting each X_i (i=1,2,...,n). - _Milan Janjic_, Jul 21 2007
%C With offset 0, a(n) is the number of ways to separate [n] into four non-overlapping intervals (allowed to be empty) and then choose a subset from each interval. - _Geoffrey Critzer_, Feb 07 2009
%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. 796.
%D Clifford A. Pickover, The Math Book, From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics, Sterling Publ., NY, 2009, page 282.
%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="/A001789/b001789.txt">Table of n, a(n) for n = 3..500</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 H. J. Brothers, <a href="http://www.brotherstechnology.com/math/pascals-prism.html">Pascal's Prism: Supplementary Material</a>.
%H Herbert Izbicki, <a href="http://www.digizeitschriften.de/dms/img/?PID=GDZPPN00247686X">Über Unterbaüme eines Baumes</a>, Monatshefte für Mathematik, Vol. 74 (1970), pp. 56-62.
%H Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>.
%H Milan Janjic and Boris Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv 1301.4550 [math.CO], 2013.
%H C. W. Jones, J. C. P. Miller, J. F. C. Conn and R. C. Pankhurst, <a href="http://dx.doi.org/10.1017/S0080454100006579">Tables of Chebyshev polynomials</a> Proc. Roy. Soc. Edinburgh. Sect. A., Vol. 62, No. 2 (1946), pp. 187-203.
%H Dusko Letic, Nenad Cakic, Branko Davidovic and Ivana Berkovic, <a href="http://www.advancesindifferenceequations.com/content/2012/1/22">Orthogonal and diagonal dimension fluxes of hyperspherical function</a>, Advances in Difference Equations 2012, 2012:22. - From _N. J. A. Sloane_, Sep 04 2012
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992, arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Hypercube.html">Hypercube</a>.
%H Alina F. Y. Zhao, <a href="http://www.emis.de/journals/JIS/VOL17/Zhao/zhao3.html">Pattern Popularity in Multiply Restricted Permutations</a>, Journal of Integer Sequences, Vol. 17 (2014), Article 14.10.3.
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-24,32,-16).
%F a(n) = 2*a(n-1) + A001788(n-2).
%F For n>0, a(n+3) = 2*A082138(n) = 8*A080930(n+1).
%F From _Paul Barry_, Apr 10 2003: (Start)
%F G.f. (with three leading zeros): x^3/(1-2*x)^4.
%F With three leading zeros, a(n) = 8*a(n-1) - 24*a(n-2) + 32*a(n-3) - 16*a(n-4), a(0)=a(1)=a(2)=0, a(3)=1.
%F E.g.f.: (x^3/3!)*exp(2*x) (with 3 leading zeros). (End)
%F a(n) = Sum_{i=3..n} binomial(i,3)*binomial(n,i). Example: for n=6, a(6) = 1*20 + 4*15 + 10*6 + 20*1 = 160. - _Bruno Berselli_, Mar 23 2018
%F From _Amiram Eldar_, Jan 06 2022: (Start)
%F Sum_{n>=3} 1/a(n) = 6*log(2) - 3.
%F Sum_{n>=3} (-1)^(n+1)/a(n) = 54*log(3/2) - 21. (End)
%p A001789:=1/(2*z-1)**4; # conjectured by _Simon Plouffe_ in his 1992 dissertation
%p seq(binomial(n+3,3)*2^n,n=0..25); # _Zerinvary Lajos_, Jun 03 2008
%t Table[Binomial[n, 3]*2^(n-3), {n,3,30}] (* _Stefan Steinerberger_, Apr 18 2006 *)
%t LinearRecurrence[{8,-24,32,-16},{1,8,40,160},30] (* _Harvey P. Dale_, Feb 10 2016 *)
%o (Haskell)
%o a001789 n = a007318 n 3 * 2 ^ (n - 3)
%o a001789_list = 1 : zipWith (+) (map (* 2) a001789_list) (drop 2 a001788_list)
%o -- _Reinhard Zumkeller_, Jul 12 2014
%o (PARI) a(n)=binomial(n,3)<<(n-3) \\ _Charles R Greathouse IV_, Sep 24 2015
%o (Magma) [Binomial(n,3)*2^(n-3): n in [3..30]]; // _G. C. Greubel_, Aug 27 2019
%o (GAP) List([3..30], n-> Binomial(n,3)*2^(n-3)); # _G. C. Greubel_, Aug 27 2019
%Y Cf. A000079, A001787, A001788, A003472, A007318, A082138, A080930.
%Y a(n) = A038207(n,3).
%K nonn,easy,nice
%O 3,2
%A _N. J. A. Sloane_
%E More terms from _James A. Sellers_, Apr 15 2000
%E More terms from _Stefan Steinerberger_, Apr 18 2006
%E Formula fixed by _Reinhard Zumkeller_, Jul 12 2014
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