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A054851 a(n) = 2^(n-7)*binomial(n,7). Number of 7D hypercubes in an n-dimensional hypercube.

%I A054851 #46 Jul 02 2025 16:01:59
%S A054851 1,16,144,960,5280,25344,109824,439296,1647360,5857280,19914752,
%T A054851 65175552,206389248,635043840,1905131520,5588385792,16066609152,
%U A054851 45364543488,126012620800,344876646400,931166945280,2483111854080
%N A054851 a(n) = 2^(n-7)*binomial(n,7). Number of 7D hypercubes in an n-dimensional hypercube.
%C A054851 If X_1,X_2,...,X_n is a partition of a 2n-set X into 2-blocks then, for n>6, a(n) is equal to the number of (n+7)-subsets of X intersecting each X_i (i=1,2,...,n). - _Milan Janjic_, Jul 21 2007
%H A054851 G. C. Greubel, <a href="/A054851/b054851.txt">Table of n, a(n) for n = 7..1000</a>
%H A054851 Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>.
%H A054851 Milan Janjic and Boris Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv:1301.4550 [math.CO], 2013.
%H A054851 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (16,-112,448,-1120,1792,-1792, 1024,-256).
%F A054851 a(n) = 2*a(n-1) + A002409(n-1).
%F A054851 a(n+8) = A082141(n+1)/2.
%F A054851 G.f.: x^7/(1-2*x)^8. - _Colin Barker_, Sep 04 2012
%F A054851 a(n) = Sum_{i=7..n} binomial(i,7)*binomial(n,i). Example: for n=11, a(11) = 1*330 + 8*165 + 36*55 + 120*11 + 330*1 = 5280. - _Bruno Berselli_, Mar 23 2018
%F A054851 From _Amiram Eldar_, Jan 06 2022: (Start)
%F A054851 Sum_{n>=7} 1/a(n) = 14*log(2) - 259/30.
%F A054851 Sum_{n>=7} (-1)^(n+1)/a(n) = 10206*log(3/2) - 124117/30. (End)
%p A054851 seq(binomial(n+7,7)*2^n,n=0..21); # _Zerinvary Lajos_, Jun 23 2008
%t A054851 Table[2^(n-7)*Binomial[n,7], {n,7,30}] (* _G. C. Greubel_, Aug 27 2019 *)
%o A054851 (PARI) vector(23, n, 2^(n-1)*binomial(n+6, 7)) \\ _G. C. Greubel_, Aug 27 2019
%o A054851 (Magma) [2^(n-7)*Binomial(n,7): n in [7..30]]; // _G. C. Greubel_, Aug 27 2019
%o A054851 (GAP) List([7..30], n-> 2^(n-7)*Binomial(n,7)); # _G. C. Greubel_, Aug 27 2019
%Y A054851 Cf. A000079, A001787, A001788, A001789, A003472, A054849, A002409.
%Y A054851 a(n) = A038207(n,7).
%K A054851 nonn,easy
%O A054851 7,2
%A A054851 _Henry Bottomley_, Apr 14 2000
%E A054851 More terms from _James Sellers_, Apr 15 2000