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%I #19 Sep 06 2018 01:52:54
%S 1,0,-1,0,4,0,-21,0,122,0,-758,0,4958,0,-33509,0,233810,0,-1641150,0,
%T 12364368,0,-71807506,0,1354944972,0,33794258600,0,2524565441138,0,
%U 186642439700891,0,16196862324254354,0,1602823227559245434,0,179707702260054046760,0
%N G.f. satisfies: inverse of function A(x) + i*A(x)^2 equals the conjugate, A(x) - i*A(x)^2, where i=sqrt(-1).
%H Paul D. Hanna, <a href="/A179270/b179270.txt">Table of n, a(n) for n = 1..256</a>
%F G.f. satisfies: A( A(x) - i*A(x)^2 ) = x*Catalan(-i*x) = i*(1-sqrt(1+4*i*x))/2.
%F a(n)=T(n,1), T(n, m)=1/2*(m/n*binomial(2*n-m-1, n-1)*(%i)^(n+m)*(-1)^n-(sum(k=m+1..n-1, T(k, m)*sum(i=k..n, T(n, i)*binomial(k, i-k)*(-%i)^(i-k)))+sum(i=m+1..n, T(n, i)*binomial(m, i-m)*(-%i)^(i-m)))), n>m, T(n,n)=1. [_Vladimir Kruchinin_, Apr 30 2012]
%e G.f.: A(x) = x - x^3 + 4*x^5 - 21*x^7 + 122*x^9 - 758*x^11 +...
%e A(x)^2 = x^2 - 2*x^4 + 9*x^6 - 50*x^8 + 302*x^10 - 1928*x^12 +...
%e A(x) + i*A(x)^2 = x - i*x^2 - x^3 + 2*i*x^4 + 4*x^5 - 9*i*x^6 - 21*x^7 - 50*i*x^8 + 122*x^9 +...
%e where Series_Reversion[A(x) + i*A(x)^2] = A(x) - i*A(x)^2.
%e The i-th iteration of A(x) + i*A(x)^2 is a real-valued series in x, and begins:
%e x - x^2 + x^3 - 2*x^5 + 3*x^6 + x^7 - 38*x^8/3 + 70*x^9/3 - 2*x^10 - 266*x^11/3 + 214*x^12 - 436*x^13/3 - 469*x^14 + 12649*x^15/9 +...
%o (PARI) {a(n)=local(A=x+sum(k=3,n-1,a(k)*x^k)+x*O(x^n));if(n==1,1,if(n%2==0,0,-polcoeff((subst(A,x,A-I*A^2)+I*subst(A,x,A-I*A^2+x*O(x^n))^2),n)/2))}
%o (PARI) /* Faster vectorized version: */
%o {oo=100;A=[1];B=x;C=(1-sqrt(1-4*(x+x^2+x*O(x^oo))))/2;A182399=[1];
%o for(n=1,oo,A182399=concat(A182399,0);B=x*Ser(A182399);
%o A182399[n]=Vec((B+subst(C+x*O(x^n),x,serreverse(B)))/2)[n];
%o A=Vec(-I*subst(x*Ser(A182399),x,I*serreverse(x+I*x^2+x^2*O(x^n))));
%o print1(A[n],", "))}
%o (Maxima)
%o T(n, m):=if n=m then 1 else 1/2*(m/n*binomial(2*n-m-1, n-1)*(%i)^(n+m)*(-1)^n-(sum(T(k, m)*sum(T(n, i)*binomial(k, i-k)*(-%i)^(i-k), i, k, n), k, m+1, n-1)+sum(T(n, i)*binomial(m, i-m)*(-%i)^(i-m), i, m+1, n)));
%o makelist(T(n, 1), n, 1, 10); [_Vladimir Kruchinin_, Apr 30 2012]
%Y Cf. A182399, A318008, A277292.
%K sign
%O 1,5
%A _Paul D. Hanna_, Jul 06 2010
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