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G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4 + x^5 * A(x/(1 - 2*x)) / (1 - 2*x).
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%I #6 Feb 16 2022 04:15:51

%S 1,1,1,1,1,1,3,9,27,81,243,731,2223,6939,22727,79971,306929,1282815,

%T 5744361,26984415,130656409,644739377,3224303841,16318576681,

%U 83717193681,436948772697,2331807007139,12791837178265,72472130039123,425239734375217,2584950704996379

%N G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4 + x^5 * A(x/(1 - 2*x)) / (1 - 2*x).

%C Shifts 5 places left under 2nd-order binomial transform.

%F a(0) = ... = a(4) = 1; a(n) = Sum_{k=0..n-5} binomial(n-5,k) * 2^k * a(n-k-5).

%t nmax = 30; A[_] = 0; Do[A[x_] = 1 + x + x^2 + x^3 + x^4 + x^5 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 < 5, 1, Sum[Binomial[n - 5, k] 2^k a[n - k - 5], {k, 0, n - 5}]]; Table[a[n], {n, 0, 30}]

%Y Cf. A004211, A007472, A010749, A210542, A275935, A351342, A351343, A351345.

%K nonn

%O 0,7

%A _Ilya Gutkovskiy_, Feb 08 2022