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%I #46 Jun 20 2023 08:22:29
%S 1,12,84,448,2016,8064,29568,101376,329472,1025024,3075072,8945664,
%T 25346048,70189056,190513152,508035072,1333592064,3451650048,
%U 8820883456,22284337152,55710842880,137950658560,338606161920
%N a(n) = 2^(n-5)*binomial(n,5). Number of 5D hypercubes in an n-dimensional hypercube.
%C With 5 leading zeros, binomial transform of binomial(n,5). - _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>4, a(n) is equal to the number of (n+5)-subsets of X intersecting each X_i (i=1,2,...,n). - _Milan Janjic_, Jul 21 2007
%H G. C. Greubel, <a href="/A054849/b054849.txt">Table of n, a(n) for n = 5..1000</a>
%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="https://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv:1301.4550 [math.CO], 2013.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (12, -60, 160, -240, 192, -64).
%F a(n) = 2*a(n-1) + A003472(n-1).
%F From _Paul Barry_, Apr 10 2003: (Start)
%F O.g.f.: x^5/(1-2*x)^6.
%F E.g.f.: exp(2*x)*(x^5/5!) (with 5 leading zeros). (End)
%F a(n) = Sum_{i=5..n} binomial(i,5)*binomial(n,i). Example: for n=8, a(8) = 1*56 + 6*28 + 21*8 + 56*1 = 448. - _Bruno Berselli_, Mar 23 2018
%F From _Amiram Eldar_, Jan 06 2022: (Start)
%F Sum_{n>=5} 1/a(n) = 10*log(2) - 35/6.
%F Sum_{n>=5} (-1)^(n+1)/a(n) = 810*log(3/2) - 655/2. (End)
%p seq(binomial(n+5,5)*2^n,n=0..22); # _Zerinvary Lajos_, Jun 13 2008
%t Table[2^(n-5)*Binomial[n,5], {n,5,30}] (* _G. C. Greubel_, Aug 27 2019 *)
%o (Sage) [lucas_number2(n, 2, 0)*binomial(n,5)/32 for n in range(5, 28)] # _Zerinvary Lajos_, Mar 10 2009
%o (PARI) vector(25, n, 2^(n-1)*binomial(n+4,5)) \\ _G. C. Greubel_, Aug 27 2019
%o (Magma) [2^(n-5)*Binomial(n,5): n in [5..30]]; // _G. C. Greubel_, Aug 27 2019
%o (GAP) List([5..30], n-> 2^(n-5)*Binomial(n,5)); # _G. C. Greubel_, Aug 27 2019
%Y Cf. A000079, A001787, A001788, A001789, A003472, A002409, A054851.
%Y a(n) = A038207(n,5).
%Y Equals 2 * A082139. First differences are in A006975.
%K easy,nonn
%O 5,2
%A _Henry Bottomley_, Apr 14 2000
%E More terms from _James A. Sellers_, Apr 15 2000
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