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%I #27 Feb 14 2024 09:17:46
%S 1,2,4,8,16,32,64,128,256,512,1024,2048,4096,8191,16369,32647,64839,
%T 127858,249528,480492,910596,1695222,3096514,5546382,9740686,16777216,
%U 28354132,47050564,76717268,123012781,194129627,301766029,462411533,699030226,1043243132
%N a(n) = Sum_{k=0..12} binomial(n,k).
%C a(n) is the number of compositions (ordered partitions) of n+1 into thirteen or fewer parts.
%C a(n) is the sum of the first thirteen terms in the n-th row of Pascal's triangle.
%H Harvey P. Dale, <a href="/A219615/b219615.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).
%F a(n) = (n^12 - 54n^11 + 1397n^10 - 21450n^9 + 218823n^8 - 1508562n^7 + 7374191n^6 - 23551110n^5 + 58206676n^4 - 48306984n^3 + 173699712n^2 + 312888960n)/479001600. - _Charles R Greathouse IV_, Nov 27 2012
%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(10)=1024, a(11)=2048, a(12)=4096, a(n)= 13*a(n-1)- 78*a(n-2)+286*a(n-3)-715*a(n-4)+1287*a(n-5)-1716*a(n-6)+ 1716*a(n-7)- 1287*a(n-8)+715*a(n-9)-286*a(n-10)+78*a(n-11)-13*a(n-12)+a(n-13). - _Harvey P. Dale_, Nov 29 2012
%e a(13)= 8191 because there are (2^13) -1 compositions of 14 into thirteen or fewer parts. When 1<= n <= 12, for n=5, a(5) = 2*a(4) = 2*16 = 32. For n=12, a(12) = 2*a(11)= 2*2048 = 4096. When n>12, for n=13, a(13) = 2*a(12) - binomial(12,12) = 2*4096 - 1 = 8191. For n = 15, a(15) = 2*a(14) - binomial(14,12) = 2*16369 - 91 = 32738 - 91 = 32647.
%t Table[Sum[Binomial[n, k], {k, 0, 12}], {n, 0, 40}] (* _T. D. Noe_, Nov 27 2012 *)
%t LinearRecurrence[{13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1},{1,2,4,8,16,32,64,128,256,512,1024,2048,4096},40] (* _Harvey P. Dale_, Nov 29 2012 *)
%o (PARI) a(n)=sum(k=1,12,binomial(n,k)) \\ _Charles R Greathouse IV_, Nov 27 2012
%Y Cf. A000127, A006261, A008859, A008860, A008861, A008862, A008863, A219531.
%K nonn,easy
%O 0,2
%A _Mokhtar Mohamed_, Nov 23 2012
%E Sequence corrected and extended by _T. D. Noe_, Nov 26 2012
%E Definition corrected by _Harvey P. Dale_, Nov 29 2012
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