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a(n) = Sum_{k=0..n} binomial(k, 6*(n-k)).
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%I #15 Apr 10 2022 13:18:38

%S 1,1,1,1,1,1,1,2,8,29,85,211,463,925,1718,3017,5097,8464,14197,24753,

%T 45697,89150,180254,368734,748924,1493990,2914906,5565127,10434412,

%U 19322901,35583926,65615746,121847272,228638698,433747259,830227401,1597653852,3078928619,5922703731,11347651254

%N a(n) = Sum_{k=0..n} binomial(k, 6*(n-k)).

%H Colin Barker, <a href="/A293169/b293169.txt">Table of n, a(n) for n = 0..1000</a>

%H V. E. Hoggatt, Jr., <a href="/A005676/a005676.pdf">7-page typed letter to N. J. A. Sloane with suggestions for new sequences</a>, circa 1977.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1,1).

%F From _Colin Barker_, Oct 17 2017: (Start)

%F G.f.: (1 - x)^5 / (1 - 6*x + 15*x^2 - 20*x^3 + 15*x^4 - 6*x^5 + x^6 - x^7).

%F a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) + a(n-7) for n>6.

%F (End)

%p f:=n-> add( binomial(k, 6*(n-k)), k=0..n);

%p [seq(f(n),n=0..30)];

%t Table[Sum[Binomial[k,6(n-k)],{k,0,n}],{n,0,40}] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1,1},{1,1,1,1,1,1,1},50] (* _Harvey P. Dale_, Apr 10 2022 *)

%o (PARI) Vec((1 - x)^5 / (1 - 6*x + 15*x^2 - 20*x^3 + 15*x^4 - 6*x^5 + x^6 - x^7) + O(x^30)) \\ _Colin Barker_, Oct 18 2017

%Y Cf. A005676, A099132.

%K nonn,easy

%O 0,8

%A _N. J. A. Sloane_, Oct 17 2017