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A380058 G.f. A(x) satisfies: A( A(x)^4 - A(x)^5 ) = x*A(x)^3.

%I A380058 #7 Jan 29 2025 12:45:48
%S A380058 1,1,2,5,13,37,111,345,1103,3603,11971,40328,137423,472829,1640328,
%T A380058 5731315,20150376,71235454,253061855,902922756,3234281950,11626416055,
%U A380058 41928973015,151656509670,550021604731,1999753822557,7287331086431,26612272636168,97375824946785,356957982413881
%N A380058 G.f. A(x) satisfies: A( A(x)^4 - A(x)^5 ) = x*A(x)^3.
%H A380058 Paul D. Hanna, <a href="/A380058/b380058.txt">Table of n, a(n) for n = 1..800</a>
%F A380058 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A380058 (1) A( A(x)^4 - A(x)^5 ) = x*A(x)^3.
%F A380058 (2) A( A(x^4 - x^5)/x^3 ) = x.
%F A380058 (3) A( x^4*A(x)^12 - x^5*A(x)^15 ) = x^3*A(x)^13 - x^3*A(x)^14.
%e A380058 G.f.: A(x) = x + x^2 + 2*x^3 + 5*x^4 + 13*x^5 + 37*x^6 + 111*x^7 + 345*x^8 + 1103*x^9 + 3603*x^10 + 11971*x^11 + 40328*x^12 + ...
%e A380058 RELATED SERIES.
%e A380058 A(x)^3 = x^3 + 3*x^4 + 9*x^5 + 28*x^6 + 87*x^7 + 276*x^8 + 893*x^9 + 2937*x^10 + ...
%e A380058 A(x)^4 = x^4 + 4*x^5 + 14*x^6 + 48*x^7 + 161*x^8 + 540*x^9 + 1822*x^10 + ...
%e A380058 A(x)^5 = x^5 + 5*x^6 + 20*x^7 + 75*x^8 + 270*x^9 + 956*x^10 + 3365*x^11 + ...
%e A380058 A(x)^4 - A(x)^5 = x^4 + 3*x^5 + 9*x^6 + 28*x^7 + 86*x^8 + 270*x^9 + 866*x^10 + 2827*x^11 + 9378*x^12 + 31538*x^13 + 107296*x^14 + 368658*x^15 + ...
%e A380058 where A( A(x)^4 - A(x)^5 ) = x*A(x)^3.
%e A380058 SPECIFIC VALUES.
%e A380058 A(t) = 1/2 at t = 0.2583423319694388182706452995079327548366116827628877452...
%e A380058   where t = 8*A(1/32).
%e A380058 A(t) = 1/3 at t = 0.22408195158505908910325809814920343873844154937613...
%e A380058   where t = 27*A(2/243).
%e A380058 A(t) = 1/4 at t = 0.18805255881224948435916264631181129351886165823425...
%e A380058   where t = 64*A(3/1024).
%e A380058 A(t) = 1/5 at t = 0.16020532597132547687124527971521135474403954761733...
%e A380058   where t = 125*A(4/3125).
%e A380058 A(t) = 1/10 at t = 0.0900081014583281267833686275811293249657866278004...
%e A380058   where t = 10^3*A(9/10^5).
%e A380058 A(1/4) = 0.42960492694042135514763101589480330859844466498925...
%e A380058   where 1/4 = A(u^4 - u^5)/u^3 with u = A(1/4).
%e A380058 A(1/5) = 0.27454979264795630599456455666046730423801890612405...
%e A380058 A(1/6) = 0.21087406049802083294286439329002681040206771579442...
%e A380058 A(1/7) = 0.17251315255186556697304716682023902930740817770490...
%e A380058 A(1/8) = 0.14637731251220491808192059768863203413840265475691...
%e A380058 A(1/32) = 0.0322927914961798522838306624384915943545764603453...
%e A380058 A(2/243) = 0.008299331540187373670491040672192719953275612939...
%o A380058 (PARI) /* From definition A( A(x)^4 - A(x)^5 ) = x*A(x)^3 */
%o A380058 {a(n) = my(A=[1, 1]); for(i=1, n, A = concat(A, 0); G=x*Ser(A); A[#A] = -Vec( subst(G, x, G^4 - G^5 )/G^2 )[#A] ); A[n]}
%o A380058 for(n=1, 40, print1(a(n), ", "))
%o A380058 (PARI) /* Using Series Reversion of A(x^4 - x^5)/x^3 */
%o A380058 {a(n) = my(A=x); for(i=1, n, A = serreverse( subst(A, x, x^4 - x^5 +x^4*O(x^n))/x^3) ); polcoeff(A, n)}
%o A380058 for(n=1, 40, print1(a(n), ", "))
%Y A380058 Cf. A268655, A380054.
%K A380058 nonn
%O A380058 1,3
%A A380058 _Paul D. Hanna_, Jan 28 2025