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a(n) = Sum_{k=0..n} binomial(9*n,9*k).
9

%I #30 Jul 14 2019 13:26:19

%S 1,2,48622,9373652,9263421862,3433541316152,2140802758677844,

%T 984101481334553024,536617781178725122150,265166261617029717011822,

%U 138567978655457801631498052,70126939586658252408697345838,36144812798331420987905742371116

%N a(n) = Sum_{k=0..n} binomial(9*n,9*k).

%H Seiichi Manyama, <a href="/A094213/b094213.txt">Table of n, a(n) for n = 0..369</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (265,139823,-6826204,-6965249,512).

%F 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 Conjectures from _Colin Barker_, May 27 2019: (Start)

%F 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 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.

%F (End)

%t Table[Sum[Binomial[9n,9k],{k,0,n}],{n,0,15}] (* _Harvey P. Dale_, Jul 14 2019 *)

%o (PARI) a(n)=sum(k=0,n,binomial(9*n,9*k))

%o (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 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 nonn,easy

%O 0,2

%A _Benoit Cloitre_, May 27 2004