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A374570 Expansion of g.f. A(x) satisfying A(x)^2 = A( A(x)*C(x) ), where C(x) = x + C(x)^2 is the Catalan function (A000108).

%I A374570 #13 Jul 13 2024 02:44:53
%S A374570 1,1,3,8,27,90,320,1152,4257,15934,60486,231894,897242,3497638,
%T A374570 13725678,54174286,214923493,856560918,3427838222,13768875142,
%U A374570 55494305328,224359469870,909656736876,3697874061870,15068978724200,61545704828266,251899370771284,1033027441769384
%N A374570 Expansion of g.f. A(x) satisfying A(x)^2 = A( A(x)*C(x) ), where C(x) = x + C(x)^2 is the Catalan function (A000108).
%C A374570 Conjecture: for n > 1, a(n) is odd iff n = 2^k + 1 for k >= 0.
%H A374570 Paul D. Hanna, <a href="/A374570/b374570.txt">Table of n, a(n) for n = 1..1030</a>
%F A374570 G.f. A(x) = Sum_{n>=1} a(n)*x^n, where B(x) = (1/x)*Series_Reversion(A(x)) is the g.f. of A374571 and C(x) = x + C(x)^2 is the g.f. of A000108, satisfies the following formulas.
%F A374570 (1) A(x)^2 = A( A(x)*C(x) ).
%F A374570 (2) x^2 = A( x*C(x*B(x)) ).
%F A374570 (3) A(x) = x / B(A(x)).
%F A374570 (4) A(x) = C(x) / B(A(x)^2).
%F A374570 (5) C(x*B(x)) = x*B(x^2).
%F A374570 (6) A(x-x^2)^2 = A( x*A(x-x^2) ).
%e A374570 G.f.: A(x) = x + x^2 + 3*x^3 + 8*x^4 + 27*x^5 + 90*x^6 + 320*x^7 + 1152*x^8 + 4257*x^9 + 15934*x^10 + 60486*x^11 + 231894*x^12 + ...
%e A374570 RELATED SERIES.
%e A374570 A(x)^2 = x^2 + 2*x^3 + 7*x^4 + 22*x^5 + 79*x^6 + 282*x^7 + 1046*x^8 + 3916*x^9 + 14907*x^10 + 57274*x^11 + 222194*x^12 + ...
%e A374570 where A(x)^2 = A( A(x)*C(x) ).
%e A374570 A(x)*C(x) = x^2 + 2*x^3 + 6*x^4 + 18*x^5 + 60*x^6 + 204*x^7 + 720*x^8 + 2586*x^9 + 9468*x^10 + 35124*x^11 + 131898*x^12 + ...
%e A374570 C(x) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + 132*x^7 + 429*x^8 + 1430*x^9 + 4862*x^10 + ... + A000108(n)*x^n + ,,,
%e A374570 where C(x) = (1 - sqrt(1 - 4*x))/2 is the Catalan function.
%e A374570 Let B(x) satisfy A(x*B(x)) = x, then
%e A374570 B(x) = 1 - x - x^2 + 2*x^3 - x^4 + x^5 + 2*x^6 - 6*x^7 - x^8 + 5*x^9 + x^10 + 2*x^12 - 8*x^13 - 6*x^14 + 22*x^15 - x^16 + ... + A374571(n)*x^n + ...
%e A374570 where C(x*B(x)) = x*B(x^2) and C(x) = x + C(x)^2.
%e A374570 Also notice that A(x-x^2) is the odd function starting as
%e A374570 A(x-x^2) = x + x^3 + 4*x^5 + 18*x^7 + 96*x^9 + 546*x^11 + 3274*x^13 + 20326*x^15 + 129622*x^17+ 843854*x^19 + ...
%e A374570 satisfying A(x-x^2)^2 = A( x*A(x-x^2) ).
%e A374570 SPECIFIC VALUES.
%e A374570 G.f. A(x) diverges at x = 1/4; what is the radius of convergence?
%e A374570 A(2/9) = 0.410501753930478190014767562028185186269192589705662553072...
%e A374570 where A(2/9)^2 = A( (1/3) * A(2/9) ).
%e A374570 A(1/5) = 0.307823207567908585715446000098072863270477544252476707540...
%e A374570 where A(1/5)^2 = A( A(1/5) * (1 - sqrt(1/5))/2 ).
%e A374570 A(1/6) = 0.222895676073964945442191376315546188067098171316653068516...
%e A374570 where A(1/6)^2 = A( A(1/6) * (1 - sqrt(1/3))/2 ).
%e A374570 A(1/8) = 0.149886223456626114071674919752683973970671151550604884301...
%e A374570 where A(1/8)^2 = A( A(1/8) * (1 - sqrt(1/2))/2 ).
%e A374570 A(1/10) = 0.11421035457722945538609562679806658343632346343476019471...
%e A374570 where A(1/10)^2 = A( A(1/10) * (1 - sqrt(3/5))/2 ).
%e A374570 A(1/12) = 0.09255115114959352826965125804331807348315032543228258146...
%e A374570 where A(1/12)^2 = A( A(1/12) * (1 - sqrt(2/3))/2 ).
%o A374570 (PARI) {a(n) = my(A=[1], Ax, C = serreverse(x-x^2 + x^2*O(x^n)));
%o A374570 for(i=1, n, A=concat(A, 0); Ax=x*Ser(A);
%o A374570 A[#A] = -polcoeff( Ax^2 - subst(Ax, x, Ax*C ), #A+1) ); A[n]}
%o A374570 for(n=1, 30, print1(a(n), ", "))
%Y A374570 Cf. A374571, A000108, A075864.
%K A374570 nonn
%O A374570 1,3
%A A374570 _Paul D. Hanna_, Jul 11 2024