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Expansion of g.f. A(x) satisfying A(x) = A( x^2*(1+x)^3 ) / x.
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%I #12 Jan 26 2024 08:33:36

%S 1,3,6,19,51,114,280,750,2064,5701,15378,40521,104947,267111,673050,

%T 1699517,4330599,11154867,29035169,76256925,201654330,535791945,

%U 1427841441,3811047033,10175838252,27152046705,72337813554,192304557673,509943239307,1348674768789,3557601342063

%N Expansion of g.f. A(x) satisfying A(x) = A( x^2*(1+x)^3 ) / x.

%C The radius of convergence r of g.f. A(x) solves r*(1+r)^3 = 1 where r = 0.3802775690976141156733...

%H Paul D. Hanna, <a href="/A369553/b369553.txt">Table of n, a(n) for n = 1..500</a>

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

%F (1) A(x) = A( x^2*(1+x)^3 ) / x.

%F (2) R(x*A(x)) = x^2*(1+x)^3, where R(A(x)) = x.

%F (3) A(x) = x * Product_{n>=1} F(n)^3, where F(1) = 1+x, and F(n+1) = 1 + (F(n) - 1)^2 * F(n)^3 for n >= 1.

%F (4) A(x) = B(x)^3/x^2 where B(x) is the g.f. of A369546.

%e G.f.: A(x) = x + 3*x^2 + 6*x^3 + 19*x^4 + 51*x^5 + 114*x^6 + 280*x^7 + 750*x^8 + 2064*x^9 + 5701*x^10 + 15378*x^11 + 40521*x^12 + ...

%e RELATED SERIES.

%e (x^2*A(x))^(1/3) = x + x^2 + x^3 + 4*x^4 + 7*x^5 + 11*x^6 + 26*x^7 + 75*x^8 + 222*x^9 + 592*x^10 + ... + A369546(n)*x^n + ...

%e Let R(x) be the series reversion of A(x),

%e R(x) = x - 3*x^2 + 12*x^3 - 64*x^4 + 399*x^5 - 2655*x^6 + 18336*x^7 - 130485*x^8 + 951780*x^9 - 7079262*x^10 + ...

%e then R(x) and g.f. A(x) satisfy:

%e (1) R(A(x)) = x,

%e (2) R(x*A(x)) = x^2*(1 + x)^3.

%e GENERATING METHOD.

%e Define F(n), a polynomial in x of order 5^(n-1), by the following recurrence:

%e F(1) = (1 + x),

%e F(2) = (1 + x^2 * (1+x)^3),

%e F(3) = (1 + x^4 * (1+x)^6 * F(2)^3),

%e F(4) = (1 + x^8 * (1+x)^12 * F(2)^6 * F(3)^3),

%e F(5) = (1 + x^16 * (1+x)^24 * F(2)^12 * F(3)^6 * F(4)^3),

%e ...

%e F(n+1) = 1 + (F(n) - 1)^2 * F(n)^3

%e ...

%e Then the g.f. A(x) equals the infinite product:

%e A(x) = x * F(1)^3 * F(2)^3 * F(3)^3 * ... * F(n)^3 * ...

%o (PARI) {a(n) = my(A=[1],F); for(i=1,n, A=concat(A,0); F=x*Ser(A); A[#A] = polcoeff( subst(F,x, x^2*(1 + x)^3 ) - x*F ,#A+1) ); A[n]}

%o for(n=1,35, print1(a(n),", "))

%Y Cf. A369546, A350432, A369552, A369554, A369555, A369556.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jan 25 2024