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%I #6 Feb 16 2022 04:15:59
%S 1,1,1,1,1,3,9,27,81,245,761,2493,8849,34519,147057,670327,3198561,
%T 15732905,79174929,407127897,2145061729,11635963499,65309080185,
%U 380583443187,2304629301041,14475031232285,93943897651017,627220447621973,4290783719133041,29988917377046207
%N G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4 * A(x/(1 - 2*x)) / (1 - 2*x).
%C Shifts 4 places left under 2nd-order binomial transform.
%F a(0) = ... = a(3) = 1; a(n) = Sum_{k=0..n-4} binomial(n-4,k) * 2^k * a(n-k-4).
%t nmax = 29; A[_] = 0; Do[A[x_] = 1 + x + x^2 + x^3 + x^4 A[x/(1 - 2 x)]/(1 - 2 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t a[n_] := a[n] = If[n < 4, 1, Sum[Binomial[n - 4, k] 2^k a[n - k - 4], {k, 0, n - 4}]]; Table[a[n], {n, 0, 29}]
%Y Cf. A004211, A007472, A010748, A210541, A275934, A351342, A351344, A351345.
%K nonn
%O 0,6
%A _Ilya Gutkovskiy_, Feb 08 2022
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