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%I #8 Nov 03 2021 05:20:29
%S 1,1,2,7,45,540,12645,578965,52968266,9592378291,3490570329073,
%T 2521575506955308,3665174976025818601,10583587128179171478201,
%U 61512603105342112799632050,710375545029057279438117199695,16513584476995892580457952423234565
%N G.f. A(x) satisfies: A(x) = 1 / (1 - x - x^2 * A(4*x)).
%F a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-2} 4^k * a(k) * a(n-k-2).
%F a(n) ~ c * 2^(n*(n-2)/2), where c = 3.18049189724646501466385558274654521200715578089919192312230814532162... - _Vaclav Kotesovec_, Nov 03 2021
%t nmax = 16; A[_] = 0; Do[A[x_] = 1/(1 - x - x^2 A[4 x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t a[0] = 1; a[n_] := a[n] = a[n - 1] + Sum[4^k a[k] a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 16}]
%Y Cf. A001006, A015085, A348878, A348879.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Nov 02 2021