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%I #7 Nov 14 2021 06:55:52
%S 1,1,9,321,42937,22259313,45726174057,374866565186721,
%T 12285883413435994137,1610409077693221284887505,
%U 844327818646575560326075164105,1770688839714867344554954935264852993,14853625190589908388648838739441430566681721
%N G.f. A(x) satisfies: A(x) = 1 / (1 - x * A(4*x)^2).
%F a(0) = 1; a(n) = Sum_{i=0..n-1} Sum_{j=0..n-i-1} 4^(i+j) * a(i) * a(j) * a(n-i-j-1).
%F a(n) ~ c * 2^(n^2), where c = 0.6660597482166910709619924328518595274303795046... - _Vaclav Kotesovec_, Nov 14 2021
%t nmax = 12; A[_] = 0; Do[A[x_] = 1/(1 - x A[4 x]^2) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t a[0] = 1; a[n_] := a[n] = Sum[Sum[4^(i + j) a[i] a[j] a[n - i - j - 1], {j, 0, n - i - 1}], {i, 0, n - 1}]; Table[a[n], {n, 0, 12}]
%Y Cf. A001764, A015085, A171192, A349286.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Nov 13 2021
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