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 A205566 G.f.: A(x) = 1 + x*A(x)*A(-x) + x^2*exp( Sum_{n>=1} 2*L(n)^2*x^(2*n)/n ), where A(x) = exp(Sum_{n>=1} L(n)*x^n/n). 1

%I #15 Mar 30 2012 18:37:34

%S 1,1,1,1,2,3,3,3,4,6,17,27,44,63,71,97,98,114,203,288,498,773,3138,

%T 5863,8018,10406,19726,32753,40372,47679,89898,154289,199094,277526,

%U 439695,742508,880786,1099183,1765472,2706524,3378956,4461436,9651388,16828143,24527676

%N G.f.: A(x) = 1 + x*A(x)*A(-x) + x^2*exp( Sum_{n>=1} 2*L(n)^2*x^(2*n)/n ), where A(x) = exp(Sum_{n>=1} L(n)*x^n/n).

%C Note: A(x)*A(-x) = exp( Sum_{n>=1} L(2*n)*x^(2*n)/n ) is a bisection of the g.f. A(x), where L(n) is the n-th coefficient in the logarithmic derivative of A(x).

%H Paul D. Hanna, <a href="/A205566/b205566.txt">Table of n, a(n) for n = 0..512</a>

%e G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 3*x^6 + 3*x^7 + 4*x^8 +...

%e Related expansions:

%e A(x)*A(-x) = 1 + x^2 + 3*x^4 + 3*x^6 + 6*x^8 + 27*x^10 + 63*x^12 + 97*x^14 + 114*x^16 + 288*x^18 + 773*x^20 + 5863*x^22 + 10406*x^24 +...

%e exp(Sum_{n>=1} 2*L(n)^2*x^(2*n)/n) = 1 + 2*x^2 + 3*x^4 + 4*x^6 + 17*x^8 + 44*x^10 + 71*x^12 + 98*x^14 + 203*x^16 + 498*x^18 + 3138*x^20 + 8018*x^22 +...

%e log(A(x)) = x + x^2/2 + x^3/3 + 5*x^4/4 + 6*x^5/5 + x^6/6 + x^7/7 + 5*x^8/8 + 10*x^9/9 + 106*x^10/10 + 111*x^11/11 +...+ L(n)*x^n/n +...

%o (PARI) {a(n)=local(L=vector(n+1,i,1),A=Ser(L));for(i=1,n,A=1+x*A*subst(A,x,-x+x*O(x^n))+x^2*exp(2*sum(m=1,#L\2,x^(2*m)*L[m]^2/m)+x*O(x^n));L=Vec(deriv(log(A))));polcoeff(A,n)}

%K nonn

%O 0,5

%A _Paul D. Hanna_, Jan 28 2012

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