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A159315 E.g.f. satisfies: d/dx log(A(x)) = A(2*x)^(1/2).

%I A159315 #8 Feb 23 2014 10:43:29
%S A159315 1,1,2,7,41,406,7127,235147,15191966,1953128401,501361942127,
%T A159315 257110692345262,263513099974512041,539923433830720468321,
%U A159315 2212048542930121133510402,18123271334339868892408048927
%N A159315 E.g.f. satisfies: d/dx log(A(x)) = A(2*x)^(1/2).
%C A159315 Row 0 of array A159314.
%H A159315 Vaclav Kotesovec, <a href="/A159315/b159315.txt">Table of n, a(n) for n = 0..78</a>
%F A159315 E.g.f. satisfies: A'(x) = A(x)*A(2*x)^(1/2).
%F A159315 a(n) = Sum_{i=0..n-1} C(n-1,i)*A126444(i)*a(n-1-i) for n>0 with a(0)=1.
%F A159315 E.g.f.: A(x) = G(x/2)^2 where G(x) = e.g.f. of A126444.
%F A159315 E.g.f.: A(x) = F(x/4)^4 where F(x) = e.g.f. of A159316.
%F A159315 a(n) ~ c * 2^(n*(n-3)/2), where c = 14.6416352593041803546... - _Vaclav Kotesovec_, Feb 23 2014
%e A159315 E.g.f.: A(x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 41*x^4/4! + 406*x^5/5! +...
%e A159315 Related expansions:
%e A159315 log(A(x)) = x +x^2/2! +3*x^3/3! +19*x^4/4! +225*x^5/5! +4801*x^6/6! +...
%e A159315 A(2*x)^(1/2) = 1 + x + 3*x^2/2! +19*x^3/3! +225*x^4/4! +4801*x^5/5! +...
%e A159315 in which the coefficients are given by A126444.
%o A159315 (PARI) {a(n)=local(A=vector(n+2, j, 1+j*x)); for(i=0, n+1, for(j=0, n, m=n+1-j; A[m]=exp(intformal((A[m+1]+x*O(x^n))^(2^(m-1)))))); n!*polcoeff(A[1], n, x)}
%Y A159315 Cf. A159314, A126444, A159316, A159317.
%K A159315 nonn
%O A159315 0,3
%A A159315 _Paul D. Hanna_, Apr 19 2009