This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A220051 #20 Feb 23 2017 15:45:42
%S A220051 1,2,4,8,16,32,64,128,256,512,1024,2048,4096,8192,16384,32767,65519,
%T A220051 130918,261156,519252,1026876,2014992,3913704,7507638,14198086,
%U A220051 26434916,48412432,87167164,154276028,268435456,459312152,773201629,1281220733,2091005866
%N A220051 Sum_{k=0..14} binomial(n,k).
%C A220051 a(n) is the number of compositions (ordered partitions) of n+1 into fifteen or fewer parts.
%C A220051 a(n) = sum(binomial(n+1,2k), for k = 0..7).
%C A220051 a(n) is the sum of the first fifteen terms in the n-th row of Pascal's triangle.
%H A220051 Indranil Ghosh, <a href="/A220051/b220051.txt">Table of n, a(n) for n = 0..1000</a>
%F A220051 a(n) = 1 + (n^14 - 77*n^13 + 2821*n^12 - 6288*n^11 + 947947*n^10 - 10081071*n^9 + 77889383*n^8 - 435638203*n^7 + 1793239448*n^6 - 5043110072*n^5 + 1111159696*n^4 - 8346754416*n^3 + 30605906304*n^2 + 57424792320*n)/14!.
%F A220051 G.f.: (1 - 13x + 79x^2 - 297x^3 + 771x^4 - 1461x^5 + 2083x^6 - 2269x^7 + 1897x^8 - 1211x^9 + 581x^10 - 203x^11 + 49x^12 - 7x^13 + x^14)/(1-x)^15.
%F A220051 a(n) = 2*a(n-1), for 1 <= n <= 14, with a(0) = 1, a(n) = 2*a(n-1) - C(n-1,14), for n> 14.
%e A220051 a(15) = 32767 because there are 2^15 = 32768 compositions of 16 into any size parts but one of the compositions (1 + 1 + ... + 1 = 16) has more than fifteen parts.
%e A220051 When 1 <= n <= 14, for n=10, a(10) = 2*a(9) = 2*512 = 1024. For n=14, a(14) = 2*a(13) = 2*8192 = 16384.
%e A220051 When n > 14, for n = 15, a(15) = 2*a(14) -C(14,14) = 2*16384 -1 = 32767. For n=20, a(20) = 2*a(19) -C(19,14) = 2*519252 -11626 = 1038504 -11626 = 1026876.
%t A220051 Table[Sum[Binomial[n,k],{k,0,14}],{n,0,33}] (* _Indranil Ghosh_, Feb 22 2017 *)
%t A220051 NestList[{#1 + 1, 2 #2 - Boole[#1 >= 14] Binomial[#1, 14]} & @@ # &, {0, 1}, 33][[All, -1]] (* _Michael De Vlieger_, Feb 22 2017 *)
%o A220051 (PARI) a(n)=sum(k=0,14,binomial(n,k)) \\ _Indranil Ghosh_, Feb 23 2017
%Y A220051 Cf. A000127, A006261, A008859, A008860, A008861, A008862, A008863, A219531, A219615, A219676.
%K A220051 nonn,easy
%O A220051 0,2
%A A220051 _Mokhtar Mohamed_, Dec 03 2012