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%I M4718 #83 Sep 08 2022 08:44:32
%S 1,10,60,280,1120,4032,13440,42240,126720,366080,1025024,2795520,
%T 7454720,19496960,50135040,127008768,317521920,784465920,1917583360,
%U 4642570240,11142168576,26528972800,62704844800,147220070400
%N a(n) = 2^(n-4)*C(n,4).
%C Number of 4D hypercubes in n-dimensional hypercube. - _Henry Bottomley_, Apr 14 2000
%C With four leading zeros, binomial transform of C(n,4). - _Paul Barry_, Apr 10 2003
%C If X_1, X_2, ..., X_n is a partition of a 2n-set X into 2-blocks, then, for n>3, a(n) is equal to the number of (n+4)-subsets of X intersecting each X_i (i=1,2,...,n). - _Milan Janjic_, Jul 21 2007
%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 and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A003472/b003472.txt">Table of n, a(n) for n = 4..400</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://resolver.sub.uni-goettingen.de/purl?GDZPPN00247686X">Über Unterbaeume eines Baumes</a>, Monatshefte für Mathematik, Vol. 74 (1970), pp. 56-62.
%H Herbert Izbicki, <a href="http://dx.doi.org/10.1007/BF01298302">Über Unterbaeume 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 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 John Riordan and N. J. A. Sloane, <a href="/A003471/a003471_1.pdf">Correspondence, 1974</a>.
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (10,-40,80,-80,32).
%F a(n) = 2*a(n-1) + A001789(n-1).
%F From _Paul Barry_, Apr 10 2003: (Start)
%F O.g.f.: x^4/(1-2*x)^5.
%F E.g.f.: exp(2*x)(x^4/4!) (with 4 leading zeros). (End)
%F a(n) = Sum_{i=4..n} binomial(i,4)*binomial(n,i). Example: for n=7, a(7) = 1*35 + 5*21 + 15*7 + 35*1 = 280. - _Bruno Berselli_, Mar 23 2018
%F From _Amiram Eldar_, Jan 06 2022: (Start)
%F Sum_{n>=4} 1/a(n) = 20/3 - 8*log(2).
%F Sum_{n>=4} (-1)^n/a(n) = 216*log(3/2) - 260/3. (End)
%p A003472:=-1/(2*z-1)**5; # conjectured by _Simon Plouffe_ in his 1992 dissertation
%p seq(binomial(n,4)*2^(n-4),n=4..24); # _Zerinvary Lajos_, Jun 12 2008
%t Table[2^(n-4) Binomial[n,4],{n,4,50}] (* or *) LinearRecurrence[{10,-40,80,-80,32},{1,10,60,280,1120},50] (* _Harvey P. Dale_, May 27 2017 *)
%o (Magma) [2^(n-4)*Binomial(n, 4): n in [4..30]]; // _Vincenzo Librandi_, Oct 16 2011
%o (PARI) a(n)=binomial(n,4)<<(n-4) \\ _Charles R Greathouse IV_, May 18 2015
%o (Sage) [2^(n-4)*binomial(n,4) for n in (4..30)] # _G. C. Greubel_, Aug 27 2019
%o (GAP) List([4..30], n-> 2^(n-4)*Binomial(n,4)); # _G. C. Greubel_, Aug 27 2019
%Y Cf. A001787, A001788, A001789.
%Y a(n) = A038207(n,4).
%K nonn,easy,nice
%O 4,2
%A _N. J. A. Sloane_
%E More terms from _James A. Sellers_, Apr 15 2000
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