A380059 - OEIS

%I #6 Jan 29 2025 12:45:55

%S 1,1,1,2,7,22,61,172,528,1696,5461,17591,57356,189786,634137,2131628,

%T 7205426,24500143,83760349,287672215,991932713,3432816902,11920342010,

%U 41521046864,145032773711,507902711094,1782900884412,6272354908197,22111650029873,78097290365451,276324399241123,979311618314876

%N G.f. A(x) satisfies: A(x)^4 = x * A( A(x)^3 + A(x)^4 ).

%H Paul D. Hanna, <a href="/A380059/b380059.txt">Table of n, a(n) for n = 1..800</a>

%F G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

%F (1) A(x)^4 = x * A( A(x)^3 + A(x)^4 ).

%F (2) A( x^4 / A(x^3 + x^4) ) = x.

%F (3) A( A(x)^12/x^3 + A(x)^16/x^4 ) = A(x)^16 / (x^4*(A(x)^3 + A(x)^4)).

%e G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 7*x^5 + 22*x^6 + 61*x^7 + 172*x^8 + 528*x^9 + 1696*x^10 + 5461*x^11 + 17591*x^12 + ...

%e RELATED SERIES.

%e A(x)^3 = x^3 + 3*x^4 + 6*x^5 + 13*x^6 + 39*x^7 + 129*x^8 + 403*x^9 + 1224*x^10 + ...

%e A(x)^4 = x^4 + 4*x^5 + 10*x^6 + 24*x^7 + 71*x^8 + 236*x^9 + 766*x^10 + ...

%e A(x)^3 + A(x)^4 = x^3 + 4*x^4 + 10*x^5 + 23*x^6 + 63*x^7 + 200*x^8 + 639*x^9 + 1990*x^10 + 6237*x^11 + 20074*x^12 + ...

%e where A( A(x)^3 + A(x)^4 ) = A(x)^4/x.

%e x^4/A(x^3 + x^4) = x - x^2 + x^3 - 2*x^4 + x^5 - x^6 + x^7 - x^8 + x^9 - 2*x^10 - x^11 - 2*x^12 - 3*x^13 - 13*x^14 - 11*x^15 - 15*x^16 - 39*x^17 - 61*x^18 + ...

%e where A( x^4/A(x^3 + x^4) ) = x.

%e SPECIFIC VALUES.

%e A(t) = 1/3 at t = 0.237617436167742897676679898250396803483311950564446...

%e   where t = (1/3)^4 / A(4/81).

%e A(t) = 1/4 at t = 0.196092137514301239531241400218586027433039522490231...

%e   where t = (1/4)^4 / A(5/256).

%e A(t) = 1/5 at t = 0.165066513409378862764418160738331339653306347173009...

%e   where t = (1/5)^4 / A(6/5^4).

%e A(t) = 1/6 at t = 0.142085514915598789873764231458595195799899204393488...

%e   where t = (1/6)^4 / A(7/6^4).

%e A(1/4) = 0.3663421370099553043795186321079524603008890521836617716...

%e   where 1/4 = u^4 / A(u^3 + u^4) with u = A(1/4).

%e A(1/5) = 0.25683998483494952191773378515674346593117123116573...

%e A(1/6) = 0.20242282088815661825831527276785970440350265885628...

%e A(1/8) = 0.14340987551783919608809673534383765997056239900000...

%e A(4/81) = 0.05195611572725837176456377031712316815548506333817...

%e A(5/256) = 0.019920482531917489082055305899615203763458473727669...

%o (PARI) /* From definition A( A(x)^3 + A(x)^4 ) = A(x)^4/x */

%o {a(n) = my(V=[0,1,1]); for(i=1, n, V = concat(V, 0); A=Ser(V); V[#V] = Vec(x*subst(A, x, A^3 + A^4)/A^4)[#V-1] ); V[n+1]}

%o for(n=1, 40, print1(a(n), ", "))

%o (PARI) /* Using Series Reversion of x^4/A(x^3 + x^4) */

%o {a(n) = my(A=x); for(i=1, n, A = serreverse( x^4/subst(A, x, x^3 + x^4 +x^3*O(x^n))) ); polcoeff(A, n)}

%o for(n=1, 40, print1(a(n), ", "))

%Y Cf. A272485.

%K nonn,new

%O 1,4

%A _Paul D. Hanna_, Jan 28 2025