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 A008862 a(n) = Sum_{k=0..9} C(n,k). 14

%I

%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} C(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

%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(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, 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). - _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 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

%O 0,2

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

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Last modified June 25 04:01 EDT 2021. Contains 345450 sequences. (Running on oeis4.)