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A002409 a(n) = 2^n*C(n+6,6). Number of 6D hypercubes in an (n+6)-dimensional hypercube.

%I A002409 M4939 N1668 #67 Jul 22 2025 07:33:36
%S A002409 1,14,112,672,3360,14784,59136,219648,768768,2562560,8200192,25346048,
%T A002409 76038144,222265344,635043840,1778122752,4889837568,13231325184,
%U A002409 35283533824,92851404800,241413652480,620777963520,1580162088960
%N A002409 a(n) = 2^n*C(n+6,6). Number of 6D hypercubes in an (n+6)-dimensional hypercube.
%C A002409 If X_1,X_2,...,X_n is a partition of a 2n-set X into 2-blocks then, for n>5, a(n-6) is equal to the number of (n+6)-subsets of X intersecting each X_i (i=1,2,...,n). - _Milan Janjic_, Jul 21 2007
%D A002409 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002409 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002409 Vincenzo Librandi, <a href="/A002409/b002409.txt">Table of n, a(n) for n = 0..400</a>
%H A002409 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 A002409 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 A002409 Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>.
%H A002409 Milan Janjic and Boris Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv:1301.4550 [math.CO], 2013.
%H A002409 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 A002409 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H A002409 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (14,-84,280,-560,672,-448,128).
%F A002409 G.f.: 1/(1-2*x)^7.
%F A002409 a(n) = 2*a(n-1) + A054849(n-1).
%F A002409 For n>0, a(n) = 2*A082140(n).
%F A002409 a(n) = Sum_{i=6..n+6} binomial(i,6)*binomial(n+6,i). Example: for n=5, a(5) = 1*462 + 7*330 + 28*165 + 84*55 + 210*11 + 462*1 = 14784. - _Bruno Berselli_, Mar 23 2018
%F A002409 From _Amiram Eldar_, Jan 06 2022: (Start)
%F A002409 Sum_{n>=0} 1/a(n) = 47/5 - 12*log(2).
%F A002409 Sum_{n>=0} (-1)^n/a(n) = 2916*log(3/2) - 5907/5. (End)
%F A002409 n*a(n) +2*(-n-6)*a(n-1)=0. - _R. J. Mathar_, Jul 22 2025
%p A002409 A002409:=-1/(2*z-1)**7; # _Simon Plouffe_ in his 1992 dissertation
%p A002409 seq(binomial(n+6,6)*2^n,n=0..22); # _Zerinvary Lajos_, Jun 16 2008
%t A002409 CoefficientList[Series[1/(1-2x)^7,{x,0,40}],x] (* or *) LinearRecurrence[ {14,-84,280,-560,672,-448,128},{1,14,112,672,3360,14784,59136},40] (* _Harvey P. Dale_, Jan 24 2022 *)
%o A002409 (Magma) [2^n*Binomial(n+6, 6): n in [0..30]]; // _Vincenzo Librandi_, Oct 14 2011
%Y A002409 First differences are in A006976.
%Y A002409 Cf. A000079, A001787, A001788, A001789, A003472, A054849, A054851, A082140.
%Y A002409 a(n) = A038207(n+6,6).
%K A002409 nonn,easy
%O A002409 0,2
%A A002409 _N. J. A. Sloane_
%E A002409 More terms from _Henry Bottomley_ and _James Sellers_, Apr 15 2000
%E A002409 Typo in definition corrected by _Zerinvary Lajos_, Jun 16 2008