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%I #15 Aug 24 2017 09:11:57
%S 1,1,4,25,204,2024,23560,315147,4777932,81249562,1536125144,
%T 32033803936,731461308600,18165906647652,487702581895824,
%U 14076885317410829,434705720055275532,14300349927653656920,499229708336982490936,18432449956359308918034,717597821570439742670440
%N G.f. satisfies: A(x) = 1 + x * sqrt( d/dx x*A(x)^4 ).
%H Paul D. Hanna, <a href="/A213192/b213192.txt">Table of n, a(n) for n = 0..300</a>
%F G.f. satisfies: A(x) = 1 + x * sqrt( A(x)^4 + 4*x*A(x)^3*A'(x) ).
%F a(m) == 1 (mod 2) at m = 2^n-1 for n>=0, otherwise a(m) == 0 (mod 2).
%F a(n) ~ c * 2^n * n! / sqrt(n), where c = 1.2412292448741911566... - _Vaclav Kotesovec_, Aug 24 2017
%e G.f.: A(x) = 1 + x + 4*x^2 + 25*x^3 + 204*x^4 + 2024*x^5 + 23560*x^6 +...
%e Related expansions:
%e A(x)^4 = 1 + 4*x + 22*x^2 + 152*x^3 + 1261*x^4 + 12252*x^5 + 137370*x^6 +...
%e d/dx x*A(x)^4 = 1 + 8*x + 66*x^2 + 608*x^3 + 6305*x^4 + 73512*x^5 +...
%e (A(x)-1)^2 = x^2 + 8*x^3 + 66*x^4 + 608*x^5 + 6305*x^6 + 73512*x^7 +...
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+x;for(i=1,n,A=1+x*deriv(x*(A+x*O(x^n))^4)^(1/2)));polcoeff(A,n)}
%o for(n=0,25,print1(a(n),", "))
%Y Cf. A226067.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Feb 28 2013
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