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A385526 E.g.f. A(x) satisfies A(x) = exp(x*A(3*x)).

%I A385526 #14 Jul 02 2025 04:18:32
%S A385526 1,1,7,208,23365,9588976,14040296659,71747056999360,
%T A385526 1255862559932597257,74168744207577385109248,
%U A385526 14599375893944236344767578111,9483024632344097320792984610415616,20158786175666520486280070249843236771213,139271933359690469686747131442731382830399594496
%N A385526 E.g.f. A(x) satisfies A(x) = exp(x*A(3*x)).
%F A385526 a(0) = 1; a(n) = Sum_{k=0..n-1} (k+1) * 3^k * binomial(n-1,k) * a(k) * a(n-1-k).
%F A385526 a(n) ~ c * n! * 3^(n*(n-1)/2), where c = 1.361839192264541770366149558100099215697354561... - _Vaclav Kotesovec_, Jul 02 2025
%t A385526 nmax = 15; A[_] = 1; Do[A[x_] = E^(x*A[3*x]) + O[x]^j // Normal, {j, 1, nmax + 1}]; CoefficientList[A[x], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Jul 02 2025 *)
%o A385526 (Ruby)
%o A385526 def ncr(n, r)
%o A385526   return 1 if r == 0
%o A385526   (n - r + 1..n).inject(:*) / (1..r).inject(:*)
%o A385526 end
%o A385526 def A(q, n)
%o A385526   ary = [1]
%o A385526   (1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + (j + 1) * q ** j * ncr(i - 1, j) * ary[j] * ary[i - 1 - j]}}
%o A385526   ary
%o A385526 end
%o A385526 def A385526(n)
%o A385526   A(3, n)
%o A385526 end
%Y A385526 Cf. A000272, A096538, A141369, A385527, A385528, A385529, A385530.
%Y A385526 Cf. A015084.
%K A385526 nonn
%O A385526 0,3
%A A385526 _Seiichi Manyama_, Jul 02 2025