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 A094213 #37 Aug 12 2025 03:46:25
%S A094213 1,2,48622,9373652,9263421862,3433541316152,2140802758677844,
%T A094213 984101481334553024,536617781178725122150,265166261617029717011822,
%U A094213 138567978655457801631498052,70126939586658252408697345838,36144812798331420987905742371116
%N A094213 a(n) = Sum_{k=0..n} binomial(9*n,9*k).
%H A094213 Seiichi Manyama, <a href="/A094213/b094213.txt">Table of n, a(n) for n = 0..369</a>
%H A094213 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (265,139823,-6826204,-6965249,512).
%F A094213 Let b(n) = a(n)-2^(9*n)/9 then b(n)+246*b(n-1)-13605*b(n-2)+b(n-3)+(-1)^n*3078=0.
%F A094213 Conjectures from _Colin Barker_, May 27 2019: (Start)
%F A094213 G.f.: (1 - 263*x - 91731*x^2 + 3035380*x^3 + 1547833*x^4) / ((1 + x)*(1 - 512*x)*(1 + 246*x - 13605*x^2 + x^3)).
%F A094213 a(n) = 265*a(n-1) + 139823*a(n-2) - 6826204*a(n-3) - 6965249*a(n-4) + 512*a(n-5) for n>4. (End)
%F A094213 a(n) ~ (1/9)*exp(n*9*log(2)) (conjecture). - _Bill McEachen_, Aug 11 2025
%t A094213 Table[Sum[Binomial[9n,9k],{k,0,n}],{n,0,15}] (* _Harvey P. Dale_, Jul 14 2019 *)
%o A094213 (PARI) a(n)=sum(k=0,n,binomial(9*n,9*k))
%o A094213 (PARI) Vec((1 - 263*x - 91731*x^2 + 3035380*x^3 + 1547833*x^4) / ((1 + x)*(1 - 512*x)*(1 + 246*x - 13605*x^2 + x^3)) + O(x^15)) \\ _Colin Barker_, May 27 2019
%Y A094213 Sum_{k=0..n} binomial(b*n,b*k): A000079 (b=1), A081294 (b=2), A007613 (b=3), A070775 (b=4), A070782 (b=5), A070967 (b=6), A094211 (b=7), A070832 (b=8), this sequence (b=9), A070833 (b=10).
%K A094213 nonn,easy
%O A094213 0,2
%A A094213 _Benoit Cloitre_, May 27 2004