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G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 - 4*x)) / (1 - 4*x).
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%I #10 Feb 20 2022 11:05:45

%S 1,1,1,5,25,129,713,4373,30289,235041,1998001,18226117,176364969,

%T 1803064033,19463340729,221691818005,2658751147297,33458500940993,

%U 440140082161121,6032572875160069,85936355674437561,1270176766188103105,19453176663852208937

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

%C Shifts 2 places left under 4th-order binomial transform.

%H Seiichi Manyama, <a href="/A351050/b351050.txt">Table of n, a(n) for n = 0..516</a>

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

%t nmax = 22; A[_] = 0; Do[A[x_] = 1 + x + x^2 A[x/(1 - 4 x)]/(1 - 4 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

%t a[0] = a[1] = 1; a[n_] := a[n] = Sum[Binomial[n - 2, k] 4^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 22}]

%Y Cf. A004213, A007472, A007476, A351049.

%K nonn

%O 0,4

%A _Ilya Gutkovskiy_, Jan 30 2022