%I #9 Oct 20 2021 07:49:44
%S 1,4,37,589,13276,386059,13741057,578451514,28109736811,1548565036789,
%T 95365652263102,6492034471389889,484086370908869821,
%U 39238367740327468444,3435176518078688461297,323029539924876486293089,32472511993953383052630556,3475005417300807667690138399
%N a(n) = 1 + 3 * Sum_{k=0..n-1} binomial(n,k) * a(k) * a(n-k-1).
%H Seiichi Manyama, <a href="/A345102/b345102.txt">Table of n, a(n) for n = 0..343</a>
%F E.g.f.: exp(x) / sqrt(7 - 6 * exp(x)).
%t a[n_] := a[n] = 1 + 3 Sum[Binomial[n, k] a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 17}]
%t nmax = 17; CoefficientList[Series[Exp[x]/Sqrt[7 - 6 Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
%t Table[Sum[Sum[Binomial[n, k] StirlingS2[k, j] 3^j (2 j - 1)!!, {j, 0, k}], {k, 0, n}], {n, 0, 17}]
%o (PARI) N=20; x='x+O('x^N); Vec(serlaplace(exp(x)/sqrt(7-6*exp(x)))) \\ _Seiichi Manyama_, Oct 20 2021
%Y Cf. A006677, A011781, A052886, A201354, A345103.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Jun 08 2021