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a(n) = Sum_{k=0..floor(n/2)} binomial(4*k,n-2*k).
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%I #9 Aug 11 2024 22:04:06

%S 1,0,1,4,7,12,30,68,137,292,644,1380,2936,6324,13625,29216,62701,

%T 134784,289547,621708,1335378,2868620,6161329,13233352,28424456,

%U 61053608,131135696,281665480,604991601,1299461088,2791106585,5995016764,12876698159,27657841516

%N a(n) = Sum_{k=0..floor(n/2)} binomial(4*k,n-2*k).

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,4,6,4,1).

%F a(n) = a(n-2) + 4*a(n-3) + 6*a(n-4) + 4*a(n-5) + a(n-6).

%F G.f.: 1/(1 - x^2*(1 + x)^4).

%o (PARI) a(n) = sum(k=0, n\2, binomial(4*k, n-2*k));

%o (PARI) my(N=40, x='x+O('x^N)); Vec(1/(1-x^2*(1+x)^4))

%Y Cf. A116090.

%K nonn,easy

%O 0,4

%A _Seiichi Manyama_, Aug 11 2024