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A385527 E.g.f. A(x) satisfies A(x) = exp(x*A(4*x)).

%I A385527 #13 Jul 02 2025 04:18:41
%S A385527 1,1,9,457,118961,152894961,940318147705,26967408304580857,
%T A385527 3534888068831469959649,2084993641133372935803249505,
%U A385527 5465706581663919414225671125834601,63043356313898446097762231466174924913065,3173076775252515207774429654590479617164788572049
%N A385527 E.g.f. A(x) satisfies A(x) = exp(x*A(4*x)).
%F A385527 a(0) = 1; a(n) = Sum_{k=0..n-1} (k+1) * 4^k * binomial(n-1,k) * a(k) * a(n-1-k).
%F A385527 a(n) ~ c * n! * 2^(n*(n-1)), where c = 1.216702003338638031273833889488221691367428313263423339843... - _Vaclav Kotesovec_, Jul 02 2025
%t A385527 nmax = 15; A[_] = 1; Do[A[x_] = E^(x*A[4*x]) + O[x]^j // Normal, {j, 1, nmax + 1}]; CoefficientList[A[x], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Jul 02 2025 *)
%o A385527 (Ruby)
%o A385527 def ncr(n, r)
%o A385527   return 1 if r == 0
%o A385527   (n - r + 1..n).inject(:*) / (1..r).inject(:*)
%o A385527 end
%o A385527 def A(q, n)
%o A385527   ary = [1]
%o A385527   (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 A385527   ary
%o A385527 end
%o A385527 def A385527(n)
%o A385527   A(4, n)
%o A385527 end
%Y A385527 Cf. A000272, A096538, A141369, A385526, A385528, A385529, A385530.
%Y A385527 Cf. A015085.
%K A385527 nonn
%O A385527 0,3
%A A385527 _Seiichi Manyama_, Jul 02 2025