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A193037 G.f. A(x) satisfies: x = Sum_{n>=1} x^n*A(-x)^A022998(n), where A022998 is defined as "if n is odd then n else 2*n.".

%I A193037 #22 Oct 18 2020 06:02:31
%S A193037 1,1,3,16,99,660,4625,33609,251024,1915365,14866307,117007587,
%T A193037 931682106,7491746385,60750081839,496214311987,4078991375519,
%U A193037 33718664525501,280123674031062,2337556609209193,19584517345276853,164677962557101656,1389268739557153255
%N A193037 G.f. A(x) satisfies: x = Sum_{n>=1} x^n*A(-x)^A022998(n), where A022998 is defined as "if n is odd then n else 2*n.".
%C A193037 Compare the g.f. to a g.f. C(x) of the Catalan numbers: x = Sum_{n>=1} x^n*C(-x)^(2*n-1).
%F A193037 G.f. satisfies: 1 = A(x)/(1 - x^2*A(x)^2) - x*A(x)^4/(1 - x^2*A(x)^4).
%F A193037 G.f. satisfies: A(x) = 1 - x^2*A(x)^2 + x*(1-x)*A(x)^4 + x^2*A(x)^5 - x^3*(1-x)*A(x)^6.
%F A193037 a(n) ~ sqrt((s^4 - 3*r^2*s^6 + 4*r^3*s^6 + 2*r*s^2*(-1 - s^2 + s^3)) / (6*s^2 - 15*r^2*s^4 + 15*r^3*s^4 + r*(-1 - 6*s^2 + 10*s^3))) / (2 * sqrt(Pi) * n^(3/2) * r^n), where r = 0.1106746599331611304198664461476598606151090027202... and s = 1.30776993974681499155955325507126073446958968382... are real roots of the system of equations 1 + r^2*s^5 + (-1 + r)*r^3*s^6 = s + r^2*s^2 + (-1 + r)*r*s^4, r^2*s^4*(5 + 6*(-1 + r)*r*s) = 1 + 2*r^2*s + 4*(-1 + r)*r*s^3. - _Vaclav Kotesovec_, Oct 18 2020
%e A193037 G.f.: A(x) = 1 + x + 3*x^2 + 16*x^3 + 99*x^4 + 660*x^5 + 4625*x^6 +...
%e A193037 The g.f. satisfies:
%e A193037 x = x*A(-x) + x^2*A(-x)^4 + x^3*A(-x)^3 + x^4*A(-x)^8 + x^5*A(-x)^5 + x^6*A(-x)^12 +...+ x^n*A(-x)^A022998(n) +...
%e A193037 where A022998 begins: [1,4,3,8,5,12,7,16,9,20,11,24,13,28,15,32,...].
%t A193037 CoefficientList[Series[Root[1 - #1 - x^2*#1^2 + (x - x^2)*#1^4 + x^2*#1^5 + (-x^3 + x^4)*#1^6 &, 1], {x, 0, 25}], x] (* _Vaclav Kotesovec_, Oct 18 2020 *)
%o A193037 (PARI) {a(n)=local(A=[1]);for(i=1,n,A=concat(A,0);A[#A]=polcoeff(sum(m=1,#A,(-x)^m*Ser(A)^((2-m%2)*m)),#A));if(n<0,0,A[n+1])}
%o A193037 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+x*A^4-x^2*(A^2+A^4-A^5)-x^3*A^6+x^4*A^6+x*O(x^n));polcoeff(A,n)}
%Y A193037 Cf. A193036, A193038, A193039, A193040, A022998.
%K A193037 nonn
%O A193037 0,3
%A A193037 _Paul D. Hanna_, Jul 14 2011