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%I #10 Jan 26 2024 08:34:01
%S 1,1,1,7,22,48,120,440,1941,8621,35496,135200,491951,1742491,6016240,
%T 20329172,68107458,229526914,786489953,2761171223,9982883073,
%U 37215348447,142623131714,558706789122,2222790569335,8927649606731,36019589692877,145421164021659,585841066191547
%N Expansion of g.f. A(x) satisfying A(x) = A( x^2*(1+x)^6 ) / (x*(1+x)^5).
%C The radius of convergence r of g.f. A(x) solves r*(1+r)^6 = 1 where r = 0.2554228710768465432050...
%H Paul D. Hanna, <a href="/A369549/b369549.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)^6 ) / (x*(1+x)^5).
%F (2) R(x*(1+x)^5*A(x)) = x^2*(1+x)^6, where R(A(x)) = x.
%F (3) A(x) = x * Product_{n>=1} F(n), where F(1) = 1+x, and F(n+1) = 1 + (F(n) - 1)^2 * F(n)^6 for n >= 1.
%F (4) A(x)^6 = x^5*B(x) where B(x) is the g.f. of A369556.
%e G.f.: A(x) = x + x^2 + x^3 + 7*x^4 + 22*x^5 + 48*x^6 + 120*x^7 + 440*x^8 + 1941*x^9 + 8621*x^10 + 35496*x^11 + 135200*x^12 + ...
%e RELATED SERIES.
%e A(x)^6/x^5 = x + 6*x^2 + 21*x^3 + 92*x^4 + 432*x^5 + 1704*x^6 + 6276*x^7 + 24096*x^8 + 95628*x^9 + ... + A369556(n)*x^n + ...
%e Let R(x) be the series reversion of A(x),
%e R(x) = x - x^2 + x^3 - 7*x^4 + 16*x^5 - 27*x^6 + 150*x^7 - 539*x^8 + 903*x^9 - 4826*x^10 + 21651*x^11 - 40859*x^12 + ...
%e then R(x) and g.f. A(x) satisfy:
%e (1) R(A(x)) = x,
%e (2) R(x*(1+x)^5*A(x)) = x^2*(1 + x)^6.
%e GENERATING METHOD.
%e Define F(n), a polynomial in x of order 8^(n-1), by the following recurrence:
%e F(1) = (1 + x),
%e F(2) = (1 + x^2 * (1+x)^6),
%e F(3) = (1 + x^4 * (1+x)^12 * F(2)^6),
%e F(4) = (1 + x^8 * (1+x)^24 * F(2)^12 * F(3)^6),
%e F(5) = (1 + x^16 * (1+x)^48 * F(2)^24 * F(3)^12 * F(4)^6),
%e ...
%e F(n+1) = 1 + (F(n) - 1)^2 * F(n)^6
%e ...
%e Then the g.f. A(x) equals the infinite product:
%e A(x) = x * F(1) * F(2) * F(3) * ... * F(n) * ...
%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)^6 ) - x*(1 + x)^5*F ,#A+1) ); A[n]}
%o for(n=1,35, print1(a(n),", "))
%Y Cf. A369556, A369545, A369546, A369547, A369548.
%K nonn
%O 1,4
%A _Paul D. Hanna_, Jan 25 2024
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