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%I #25 Jun 30 2015 17:30:59
%S 1,2,18,244,4090,78636,1670452,38369256,939975210,24332874300,
%T 661203162972,18767610621144,554355352005156,16990196093594808,
%U 539037630578944680,17669190941910393296,597443704506079947274,20809975126591414324188,745817743891030392909644,27474840405379231845795960
%N G.f. satisfies: A(x) = Series_Reversion( x - 2*A(x)*B(x) ), where B'(x) = 2*A(x).
%C Conjecture: n | a(n) for n>=1 and therefore all the terms of this sequence are integers.
%H Paul D. Hanna, <a href="/A259270/b259270.txt">Table of n, a(n) for n = 1..200</a>
%F Let B(x) = Integral 2*A(x) dx, then g.f. A(x) satisfies:
%F (1) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) 2^n * A(x)^n * B(x)^n / n!.
%F (2) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) 2^n * A(x)^n * B(x)^n / (n!*x) ).
%F a(n)/n = A259271(n).
%e G.f.: A(x) = x + 2*x^3 + 18*x^5 + 244*x^7 + 4090*x^9 + 78636*x^11 +...+ a(n)*x^(2*n-1) +...
%e Let B(x) = Integral 2*A(x) dx, an integer series that begins:
%e B(x) = x^2 + x^4 + 6*x^6 + 61*x^8 + 818*x^10 + 13106*x^12 + 238636*x^14 + 4796157*x^16 + 104441690*x^18 +...+ A259271(n)*x^(2*n) +...
%e then A(x - 2*A(x)*B(x)) = x.
%e Also,
%e A(x) = x + 2*A(x)*B(x) + [d/dx 4*A(x)^2*B(x)^2]/2! + [d^2/dx^2 8*A(x)^3*B(x)^3]/3! + [d^3/dx^3 16*A(x)^4*B(x)^4]/4! + [d^4/dx^4 32*A(x)^5*B(x)^5]/5! +...
%e Logarithmic series:
%e log(A(x)/x) = 2*A(x)*B(x)/x + [d/dx 4*A(x)^2*B(x)^2/x]/2! + [d^2/dx^2 8*A(x)^3*B(x)^3/x]/3! + [d^3/dx^3 16*A(x)^4*B(x)^4/x]/4! + [d^4/dx^4 32*A(x)^5*B(x)^5/x]/5! +...
%e Explicitly,
%e log(A(x)/x) = 4*x^2/2 + 64*x^4/4 + 1264*x^6/6 + 28064*x^8/8 + 675504*x^10/10 + 17304544*x^12/12 + 466669536*x^14/14 +...+ A259272(n)*x^(2*n)/(2*n) +...
%o (PARI) {a(n)=local(A=x+x*O(x^n),B=x^2); for(i=1, n, B=intformal(2*A); A = serreverse(x - 2*A*B +O(x^(2*n)))); polcoeff(A, 2*n-1)}
%o for(n=1, 25, print1(a(n), ", "))
%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o {a(n)=local(A=x+x*O(x^n),B=x^2); for(i=1, n, B=intformal(2*A); A = x + sum(m=1, n+1, Dx(m-1, 2^m*A^m*B^m/m!)) +O(x^(2*n))); polcoeff(A, 2*n-1)}
%o for(n=1, 25, print1(a(n), ", "))
%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o {a(n)=local(A=x+x*O(x^n),B=x^2); for(i=1, n, B=intformal(2*A); A = x*exp(sum(m=1, n, Dx(m-1, 2^m*A^m*B^m/(m!*x))) +O(x^(2*n)))); polcoeff(A, 2*n-1)}
%o for(n=1, 25, print1(a(n), ", "))
%Y Cf. A259271, A259272, A259268.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jun 29 2015
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