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A301933 G.f. A(x) satisfies: A(x) = x*(1 + 4*A(x)*A'(x)) / (1 + A(x)*A'(x)).

%I A301933 #13 Oct 14 2020 08:46:02
%S A301933 1,3,24,291,4596,88230,1979088,50570823,1446341388,45706515546,
%T A301933 1580322048288,59318131995822,2401809350808552,104347127373249036,
%U A301933 4842030589556434656,239028273094016840223,12508863342589554285372,691783629316556340447570,40316336264435949765811968
%N A301933 G.f. A(x) satisfies: A(x) = x*(1 + 4*A(x)*A'(x)) / (1 + A(x)*A'(x)).
%C A301933 Compare to: C(x) = x*(1 + 2*C(x)*C'(x)) / (1 + C(x)*C'(x)) holds when C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).
%C A301933 a(n = 2^k) is odd for k>=0, and a(n) is even elsewhere (conjecture).
%H A301933 Paul D. Hanna, <a href="/A301933/b301933.txt">Table of n, a(n) for n = 1..400</a>
%F A301933 a(n) ~ c * 3^n * n! * n^(1/3), where c = 0.113581779257198505098700336... - _Vaclav Kotesovec_, Oct 14 2020
%e A301933 G.f.: A(x) = x + 3*x^2 + 24*x^3 + 291*x^4 + 4596*x^5 + 88230*x^6 + 1979088*x^7 + 50570823*x^8 + 1446341388*x^9 + 45706515546*x^10 + ...
%e A301933 such that A = A(x) satisfies: A = x*(1 + 4*A*A')/(1 + A*A').
%e A301933 Odd coefficients in A(x) seem to occur only for x^(2^k), k>=0.
%e A301933 RELATED SERIES.
%e A301933 A(x)*A'(x) = x + 9*x^2 + 114*x^3 + 1815*x^4 + 34542*x^5 + 763014*x^6 + 19171380*x^7 + 539667387*x^8 + 16817885070*x^9 + 574647250650*x^10 + ...
%e A301933 Odd coefficients in A(x)*A'(x) also seem to occur only for x^(2^k), k>=0.
%o A301933 (PARI) {a(n) = my(L=x); for(i=1,n, L = x*(1 + 4*L'*L)/(1 + L'*L +x*O(x^n)) ); polcoeff(L,n)}
%o A301933 for(n=1,30,print1(a(n),", "))
%Y A301933 Cf. A301930, A301931, A301932.
%K A301933 nonn
%O A301933 1,2
%A A301933 _Paul D. Hanna_, Mar 28 2018