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A352236 G.f. A(x) satisfies: A(x) = 1 + x*A(x)^2 / (A(x) - 2*x*A'(x)).

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