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A008861 a(n) = Sum_{k=0..8} binomial(n,k). 14

%I #35 Jun 24 2023 13:24:54

%S 1,2,4,8,16,32,64,128,256,511,1013,1981,3797,7099,12911,22819,39203,

%T 65536,106762,169766,263950,401930,600370,880970,1271626,1807781,

%U 2533987,3505699,4791323,6474541,8656937,11460949,15033173,19548046

%N a(n) = Sum_{k=0..8} binomial(n,k).

%C a(n) is the number of compositions (ordered partitions) of n+1 into nine or fewer parts. - _Geoffrey Critzer_, Jan 24 2009

%C a(n) is the maximal number of regions in 8-space formed by n-1 7-dimensional hypercubes. Also the number of binary words of length n matching the regular expression 0*1*0*1*0*1*0*1*0*. A000124, A000125, A000127, A006261, A008859, A008860 count binary words of the form 0*1*0*, 1*0*1*0*, 0*1*0*1*0*, 1*0*1*0*1*0*, 0*1*0*1*0*1*0*, and 1*0*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="/A008861/b008861.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).

%F a(n) = Sum_{k=0..4} binomial(n+1, 2*k), compare A008859.

%F From _Geoffrey Critzer_, Jan 24 2009: (Start)

%F G.f.: (1 - 7*x + 22*x^2 - 40*x^3 + 46*x^4 - 34*x^5 + 16*x^6 - 4*x^7 + x^8)/(1-x)^9.

%F a(n) = (n^8 - 20*n^7 + 210*n^6 - 1064*n^5 + 3969*n^4 - 4340*n^3 + 15980*n^2 + 25584*n + 40320)/8!. (End)

%e a(9)=511 because all but one (namely 1+1+1+...+1=10) of the 2^9 compositions of 10 are in nine or fewer parts. - _Geoffrey Critzer_, Jan 24 2009

%p seq(sum(binomial(n,j), j=0..8), n=0..40); # _G. C. Greubel_, Sep 13 2019

%t Sum[Binomial[Range[41]-1, j-1], {j,9}] (* _G. C. Greubel_, Sep 13 2019 *)

%o (Haskell)

%o a008861 = sum . take 9 . a007318_row -- _Reinhard Zumkeller_, Nov 24 2012

%o (PARI) vector(40, n, sum(j=0,8, binomial(n-1,j))) \\ _G. C. Greubel_, Sep 13 2019

%o (Magma) [(&+[Binomial(n,k): k in [0..8]]): n in [0..40]]; // _G. C. Greubel_, Sep 13 2019

%o (Sage) [sum(binomial(n,k) for k in (0..8)) for n in (0..40)] # _G. C. Greubel_, Sep 13 2019

%o (GAP) List([0..40], n-> Sum([0..8], k-> Binomial(n,k)) ); # _G. C. Greubel_, Sep 13 2019

%Y Cf. A008859, A008860, A008862, A008863, A006261, A000127.

%Y Cf. A007318, A219531.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, _R. K. Guy_

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Last modified March 29 07:27 EDT 2024. Contains 371265 sequences. (Running on oeis4.)