%I #12 Nov 16 2023 12:53:42
%S 1,1,3,19,185,2353,36075,638115,12683761,278485217,6674259667,
%T 173097575603,4826128088489,143896870347793,4568544366818747,
%U 153883892657000259,5481761893234193889,205939077652874352577,8138639816942009694627,337568614331296733526867
%N G.f. A(x) satisfies: A(x) = 1 + x*A(x)^2 / (A(x) - 2*x*A'(x)).
%H Vaclav Kotesovec, <a href="/A352236/b352236.txt">Table of n, a(n) for n = 0..400</a>
%F G.f. A(x) satisfies:
%F (1) [x^n] A(x)^(2*n+1) = [x^(n-1)] (2*n+1) * A(x)^(2*n+1) for n >= 1.
%F (2) A(x) = 1 + x*A(x)^2/(A(x) - 2*x*A'(x)).
%F (3) A'(x) = A(x) * (1 + x*A(x)/(1 - A(x))) / (2*x).
%F (4) A(x) = exp( Integral (1 + x*A(x)/(1 - A(x)))/(2*x) dx ).
%F a(n) ~ c * 2^n * n! * n^(3/2), where c = 0.06926688933886004638602492... - _Vaclav Kotesovec_, Nov 16 2023
%e G.f.: A(x) = 1 + x + 3*x^2 + 19*x^3 + 185*x^4 + 2353*x^5 + 36075*x^6 + 638115*x^7 + 12683761*x^8 + ...
%e such that A(x) = 1 + x*A(x)^2/(A(x) - 2*x*A'(x)).
%e Related table.
%e The table of coefficients of x^k in A(x)^(2*n+1) begins:
%e n=0: [1, 1, 3, 19, 185, 2353, 36075, ...];
%e n=1: [1, 3, 12, 76, 705, 8595, 127680, ...];
%e n=2: [1, 5, 25, 165, 1490, 17506, 252050, ...];
%e n=3: [1, 7, 42, 294, 2632, 30016, 419454, ...];
%e n=4: [1, 9, 63, 471, 4239, 47295, 643017, ...];
%e n=5: [1, 11, 88, 704, 6435, 70785, 939312, ...];
%e n=6: [1, 13, 117, 1001, 9360, 102232, 1329016, ...]; ...
%e in which the following pattern holds:
%e [x^n] A(x)^(2*n+1) = [x^(n-1)] (2*n+1) * A(x)^(2*n+1), n >= 1,
%e as illustrated by
%e [x^1] A(x)^3 = 3 = [x^0] 3*A(x)^3 = 3*1;
%e [x^2] A(x)^5 = 25 = [x^1] 5*A(x)^5 = 5*5;
%e [x^3] A(x)^7 = 294 = [x^2] 7*A(x)^7 = 7*42;
%e [x^4] A(x)^9 = 4239 = [x^3] 9*A(x)^9 = 9*471;
%e [x^5] A(x)^11 = 70785 = [x^4] 11*A(x)^11 = 11*6435;
%e [x^6] A(x)^13 = 1329016 = [x^5] 13*A(x)^13 = 13*102232; ...
%e Also, compare the above terms along the diagonal to the series
%e B(x) = A(x*B(x)^2) = 1 + x + 5*x^2 + 42*x^3 + 471*x^4 + 6435*x^5 + 102232*x^6 + 1837630*x^7 + ... + A317352(n)*x^n + ...
%e where B(x)^2 = (1/x) * Series_Reversion( x/A(x)^2 ).
%o (PARI) /* Using A(x) = 1 + x*A(x)^2/(A(x) - 2*x*A'(x)) */
%o {a(n) = my(A=1); for(i=1,n, A = 1 + x*A^2/(A - 2*x*A' + x*O(x^n)) );
%o polcoeff(A,n)}
%o for(n=0,30, print1(a(n),", "))
%o (PARI) /* Using [x^n] A(x)^(2*n+1) = [x^(n-1)] (2*n+1)*A(x)^(2*n+1) */
%o {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
%o A[#A] = polcoeff((x*Ser(A)^(2*(#A)-1) - Ser(A)^(2*(#A)-1)/(2*(#A)-1)),#A-1));A[n+1]}
%o for(n=0,30, print1(a(n),", "))
%Y Cf. A088715, A286797, A317352, A352235, A352237, A352238.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Mar 08 2022