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%I #57 Jun 24 2023 13:24:15
%S 1,2,4,8,16,32,64,127,247,466,848,1486,2510,4096,6476,9949,14893,
%T 21778,31180,43796,60460,82160,110056,145499,190051,245506,313912,
%U 397594,499178,621616,768212,942649,1149017,1391842,1676116,2007328
%N a(n) = Sum_{k=0..6} binomial(n,k).
%C a(n) is the maximal number of regions in 6-space formed by n-1 5-dimensional hypercubes. - _Christian Schroeder_, Jan 04 2016
%C a(n) is the number of binary words of length n matching the regular expression 0*1*0*1*0*1*0*. A000124, A000125, A000127, A006261 count binary words of the form 0*1*0*, 1*0*1*0*, 0*1*0*1*0*, and 1*0*1*0*1*0*, respectively. - _Manfred Scheucher_, Jun 22 2023
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 72, Problem 2.
%H Reinhard Zumkeller, <a href="/A008859/b008859.txt">Table of n, a(n) for n = 0..10000</a>
%H Ângela Mestre and José Agapito, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Mestre/mestre2.html">Square Matrices Generated by Sequences of Riordan Arrays</a>, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F a(n) = Sum_{k=0..3} binomial(n+1, 2*k). - _Len Smiley_, Oct 20 2001
%F O.g.f.: (1 - 5*x + 11*x^2 - 13*x^3 + 9*x^4 - 3*x^5 + x^6)/(1-x)^7. - _R. J. Mathar_, Apr 02 2008
%F a(n) = a(n-1) + A006261(n-1). - _Christian Schroeder_, Jan 04 2016
%F a(n) = (n^6 - 9*n^5 + 55*n^4 - 75*n^3 + 304*n^2 + 444*n + 720)/720. - _Gerry Martens_ , May 04 2016
%F E.g.f.: (720 + 720*x + 360*x^2 + 120*x^3 + 30*x^4 + 6*x^5 + x^6)*exp(x)/6!. - _Ilya Gutkovskiy_, May 04 2016
%p A008859 := proc(n)
%p add(binomial(n,k),k=0..6) ;
%p end proc: # _R. J. Mathar_, Oct 30 2015
%t Table[Sum[Binomial[n,k],{k,0,6}],{n,0,40}] (* _Harvey P. Dale_, Jan 16 2012 *)
%o (Haskell)
%o a008859 = sum . take 7 . a007318_row -- _Reinhard Zumkeller_, Nov 24 2012
%o (PARI) a(n)=sum(k=0,6,binomial(n,k)) \\ _Charles R Greathouse IV_, Sep 24 2015
%o (Magma) [(&+[Binomial(n,k): k in [0..6]]): n in [0..40]]; // _G. C. Greubel_, Sep 13 2019
%o (Sage) [sum(binomial(n,k) for k in (0..6)) for n in (0..40)] # _G. C. Greubel_, Sep 13 2019
%o (GAP) List([0..40], n-> Sum([0..6], k-> Binomial(n,k)) ); # _G. C. Greubel_, Sep 13 2019
%Y Cf. A008860, A008861, A008862, A008863, A006261, A000127, A007318, A219531.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_ and _R. K. Guy_
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