The OEIS is supported by the many generous donors to the OEIS Foundation.
%I #5 Sep 10 2024 00:22:50
%S 1,1,6,65,994,19386,456940,12594465,396969930,14078044862,
%T 554782989908,24053551260186,1138039204281236,58353983394380500,
%U 3223791843357228120,190914111715994215905,12065701995815379444954,810602692757305194731094,57688894099612173692496580
%N a(0) = 1; a(n) = Sum_{k=0..n-1} (4*k+1) * a(k) * a(n-k-1).
%F G.f. A(x) satisfies: A(x) = 1 + x * A(x)^2 + 4 * x^2 * A'(x) * A(x).
%t a[0] = 1; a[n_] := a[n] = Sum[(4 k + 1) a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
%t nmax = 18; A[_] = 0; Do[A[x_] = 1 + x A[x]^2 + 4 x^2 A'[x] A[x] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%Y Cf. A000699, A088716, A151403, A215648, A375393, A376086.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Sep 09 2024