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%I #10 Jan 26 2024 08:33:40
%S 1,4,10,36,123,344,976,3000,9505,30572,98478,313644,985093,3044616,
%T 9258732,27861672,83564737,251019564,757389494,2299300236,7026093837,
%U 21596604824,66699264412,206746396728,642598368442,2001293609760,6241460893404,19481739013312,60829706955774,189911135160648
%N Expansion of g.f. A(x) satisfying A(x) = A( x^2*(1+x)^4 ) / x.
%C The radius of convergence r of g.f. A(x) solves r*(1+r)^4 = 1 where r = 0.32471795724474602596...
%H Paul D. Hanna, <a href="/A369554/b369554.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)^4 ) / x.
%F (2) R(x*A(x)) = x^2*(1+x)^4, where R(A(x)) = x.
%F (3) A(x) = x * Product_{n>=1} F(n)^4, where F(1) = 1+x, and F(n+1) = 1 + (F(n) - 1)^2 * F(n)^4 for n >= 1.
%F (4) A(x) = B(x)^4/x^3 where B(x) is the g.f. of A369547.
%e G.f.: A(x) = x + 4*x^2 + 10*x^3 + 36*x^4 + 123*x^5 + 344*x^6 + 976*x^7 + 3000*x^8 + 9505*x^9 + 30572*x^10 + 98478*x^11 + 313644*x^12 + ...
%e RELATED SERIES.
%e (x^3*A(x))^(1/4) = x + x^2 + x^3 + 5*x^4 + 11*x^5 + 19*x^6 + 46*x^7 + 150*x^8 + 527*x^9 + 1743*x^10 + ... + A369547(n)*x^n + ...
%e Let R(x) be the series reversion of A(x),
%e R(x) = x - 4*x^2 + 22*x^3 - 156*x^4 + 1265*x^5 - 10956*x^6 + 98880*x^7 - 920508*x^8 + 8779768*x^9 - 85360608*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)^4.
%e GENERATING METHOD.
%e Define F(n), a polynomial in x of order 6^(n-1), by the following recurrence:
%e F(1) = (1 + x),
%e F(2) = (1 + x^2 * (1+x)^4),
%e F(3) = (1 + x^4 * (1+x)^8 * F(2)^4),
%e F(4) = (1 + x^8 * (1+x)^16 * F(2)^8 * F(3)^4),
%e F(5) = (1 + x^16 * (1+x)^32 * F(2)^16 * F(3)^8 * F(4)^4),
%e ...
%e F(n+1) = 1 + (F(n) - 1)^2 * F(n)^4
%e ...
%e Then the g.f. A(x) equals the infinite product:
%e A(x) = x * F(1)^4 * F(2)^4 * F(3)^4 * ... * F(n)^4 * ...
%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)^4 ) - x*F ,#A+1) ); A[n]}
%o for(n=1,35, print1(a(n),", "))
%Y Cf. A369547, A350432, A369552, A369553, A369555, A369556.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jan 25 2024
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