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G.f. A(x) satisfies: A(x) = 1 + x * A(x/(1 - 2*x)) / (1 - 2*x)^2.
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%I #6 Feb 19 2022 13:54:37

%S 1,1,5,23,119,709,4749,35031,281271,2438565,22673021,224739303,

%T 2363075191,26246762213,306830932749,3763323446487,48292462190743,

%U 646763208308421,9020009372203965,130737162573013159,1965798562640921879,30613694640191725381

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

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

%t nmax = 21; A[_] = 0; Do[A[x_] = 1 + x A[x/(1 - 2 x)]/(1 - 2 x)^2 + O[x]^(nmax + 1) // Normal,nmax + 1]; CoefficientList[A[x], x]

%t a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k - 1] 2^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]

%Y Cf. A004211, A040027, A122704, A351757.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Feb 18 2022