|
%I #41 Jun 24 2023 13:25:06
%S 1,2,4,8,16,32,64,128,256,512,1023,2036,4017,7814,14913,27824,50643,
%T 89846,155382,262144,431910,695860,1097790,1698160,2579130,3850756,
%U 5658537,8192524,11698223,16489546,22964087,31621024,43081973,58115146,77663192,102875128
%N a(n) = Sum_{k=0..9} binomial(n,k).
%C a(n) is the number of compositions (ordered partitions) of n+1 into ten or fewer parts. - _Geoffrey Critzer_, Jan 24 2009
%C a(n) is the maximal number of regions in 9-space formed by n-1 8-dimensional hypercubes. Also the number of binary words of length n matching the regular expression 1*0*1*0*1*0*1*0*1*0*. A000124, A000125, A000127, A006261, A008859, A008860, A008861 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*, 1*0*1*0*1*0*1*0* and 0*1*0*1*0*1*0*1*0* respectively. - _Manfred Scheucher_, Jun 23 2023
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 72, Problem 2.
%H Reinhard Zumkeller, <a href="/A008862/b008862.txt">Table of n, a(n) for n = 0..10000</a>
%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
%F a(n) = Sum_{k=1..5} binomial(n+1, 2*k-1), compare A008860.
%F From _Geoffrey Critzer_, Jan 24 2009: (Start)
%F G.f.: (1 - 8*x + 29*x^2 - 62*x^3 + 86*x^4 - 80*x^5 + 50*x^6 - 20*x^7 + 5*x^8)/(1-x)^10.
%F a(n) = (n^9 - 27*n^8 + 366*n^7 - 2646*n^6 + 12873*n^5 - 31563*n^4 + 79064*n^3 + 34236*n^2 + 270576*n + 362880)/9!. (End)
%F a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10); a(0)=1, a(1)=2, a(2)=4, a(3)=8, a(4)=16, a(5)=32, a(6)=64, a(7)=128, a(8)=256, a(9)=512. - _Harvey P. Dale_, Mar 18 2012
%e a(10)=1023 because there are (2^10)-1 compositions of 11 into ten or fewer parts. - _Geoffrey Critzer_, Jan 24 2009
%p seq(add(binomial(n,j), j=0..9), n=0..40); # _G. C. Greubel_, Sep 13 2019
%t Table[Sum[Binomial[n,k],{k,0,9}],{n,0,40}] (* or *) LinearRecurrence[ {10,-45,120,-210,252,-210,120,-45,10,-1}, {1,2,4,8,16,32,64,128,256, 512}, 40] (* _Harvey P. Dale_, Mar 18 2012 *)
%o (Haskell)
%o a008862 = sum . take 10 . a007318_row -- _Reinhard Zumkeller_, Nov 24 2012
%o (PARI) vector(40, n, sum(j=0,9, binomial(n-1,j))) \\ _G. C. Greubel_, Sep 13 2019
%o (Magma) [(&+[Binomial(n,k): k in [0..9]]): n in [0..40]]; // _G. C. Greubel_, Sep 13 2019
%o (Sage) [sum(binomial(n,k) for k in (0..9)) for n in (0..40)] # _G. C. Greubel_, Sep 13 2019
%o (GAP) List([0..40], n-> Sum([0..9], k-> Binomial(n,k)) ); # _G. C. Greubel_, Sep 13 2019
%Y Cf. A008859, A008860, A008861, A008863, A006261, A000127.
%Y Cf. A007318, A219531.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, _R. K. Guy_
|
