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A369554
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Expansion of g.f. A(x) satisfying A(x) = A( x^2*(1+x)^4 ) / x.
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7
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1, 4, 10, 36, 123, 344, 976, 3000, 9505, 30572, 98478, 313644, 985093, 3044616, 9258732, 27861672, 83564737, 251019564, 757389494, 2299300236, 7026093837, 21596604824, 66699264412, 206746396728, 642598368442, 2001293609760, 6241460893404, 19481739013312, 60829706955774, 189911135160648
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OFFSET
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1,2
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COMMENTS
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The radius of convergence r of g.f. A(x) solves r*(1+r)^4 = 1 where r = 0.32471795724474602596...
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LINKS
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FORMULA
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G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies:
(1) A(x) = A( x^2*(1+x)^4 ) / x.
(2) R(x*A(x)) = x^2*(1+x)^4, where R(A(x)) = x.
(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.
(4) A(x) = B(x)^4/x^3 where B(x) is the g.f. of A369547.
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EXAMPLE
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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 + ...
RELATED SERIES.
(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 + ...
Let R(x) be the series reversion of A(x),
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 + ...
then R(x) and g.f. A(x) satisfy:
(1) R(A(x)) = x,
(2) R(x*A(x)) = x^2*(1 + x)^4.
GENERATING METHOD.
Define F(n), a polynomial in x of order 6^(n-1), by the following recurrence:
F(1) = (1 + x),
F(2) = (1 + x^2 * (1+x)^4),
F(3) = (1 + x^4 * (1+x)^8 * F(2)^4),
F(4) = (1 + x^8 * (1+x)^16 * F(2)^8 * F(3)^4),
F(5) = (1 + x^16 * (1+x)^32 * F(2)^16 * F(3)^8 * F(4)^4),
...
F(n+1) = 1 + (F(n) - 1)^2 * F(n)^4
...
Then the g.f. A(x) equals the infinite product:
A(x) = x * F(1)^4 * F(2)^4 * F(3)^4 * ... * F(n)^4 * ...
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PROG
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(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]}
for(n=1, 35, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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