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A369548
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Expansion of g.f. A(x) satisfying A(x) = A( x^2*(1+x)^5 ) / (x*(1+x)^4).
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7
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1, 1, 1, 6, 16, 31, 76, 267, 1067, 4158, 14773, 48624, 154124, 477235, 1444512, 4319533, 13001635, 39984251, 126648990, 414577918, 1401344342, 4867242685, 17246577931, 61892834305, 223556783337, 808761979971, 2919883601666, 10493659868180, 37480931951560, 132938081422654
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OFFSET
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1,4
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COMMENTS
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The radius of convergence r of g.f. A(x) solves r*(1+r)^5 = 1 where r = 0.2851990332453493679...
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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)^5 ) / (x*(1+x)^4).
(2) R(x*(1+x)^4*A(x)) = x^2*(1+x)^5, where R(A(x)) = x.
(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)^5 for n >= 1.
(4) A(x)^5 = x^4*B(x) where B(x) is the g.f. of A369555.
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EXAMPLE
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G.f.: A(x) = x + x^2 + x^3 + 6*x^4 + 16*x^5 + 31*x^6 + 76*x^7 + 267*x^8 + 1067*x^9 + 4158*x^10 + 14773*x^11 + 48624*x^12 + ...
RELATED SERIES.
A(x)^5/x^4 = x + 5*x^2 + 15*x^3 + 60*x^4 + 245*x^5 + 826*x^6 + 2685*x^7 + 9285*x^8 + 33170*x^9 + ... + A369555(n)*x^n + ...
Let R(x) be the series reversion of A(x),
R(x) = x - x^2 + x^3 - 6*x^4 + 16*x^5 - 31*x^6 + 126*x^7 - 492*x^8 + 1180*x^9 - 4092*x^10 + 17972*x^11 - 52498*x^12 + ...
then R(x) and g.f. A(x) satisfy:
(1) R(A(x)) = x,
(2) R(x*(1+x)^4*A(x)) = x^2*(1 + x)^5.
GENERATING METHOD.
Define F(n), a polynomial in x of order 7^(n-1), by the following recurrence:
F(1) = (1 + x),
F(2) = (1 + x^2 * (1+x)^5),
F(3) = (1 + x^4 * (1+x)^10 * F(2)^5),
F(4) = (1 + x^8 * (1+x)^20 * F(2)^10 * F(3)^5),
F(5) = (1 + x^16 * (1+x)^40 * F(2)^20 * F(3)^10 * F(4)^5),
...
F(n+1) = 1 + (F(n) - 1)^2 * F(n)^5
...
Then the g.f. A(x) equals the infinite product:
A(x) = x * F(1) * F(2) * F(3) * ... * F(n) * ...
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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)^5 ) - x*(1 + x)^4*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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