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A369553
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Expansion of g.f. A(x) satisfying A(x) = A( x^2*(1+x)^3 ) / x.
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7
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1, 3, 6, 19, 51, 114, 280, 750, 2064, 5701, 15378, 40521, 104947, 267111, 673050, 1699517, 4330599, 11154867, 29035169, 76256925, 201654330, 535791945, 1427841441, 3811047033, 10175838252, 27152046705, 72337813554, 192304557673, 509943239307, 1348674768789, 3557601342063
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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)^3 = 1 where r = 0.3802775690976141156733...
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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)^3 ) / x.
(2) R(x*A(x)) = x^2*(1+x)^3, where R(A(x)) = x.
(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.
(4) A(x) = B(x)^3/x^2 where B(x) is the g.f. of A369546.
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EXAMPLE
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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 + ...
RELATED SERIES.
(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 + ...
Let R(x) be the series reversion of A(x),
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 + ...
then R(x) and g.f. A(x) satisfy:
(1) R(A(x)) = x,
(2) R(x*A(x)) = x^2*(1 + x)^3.
GENERATING METHOD.
Define F(n), a polynomial in x of order 5^(n-1), by the following recurrence:
F(1) = (1 + x),
F(2) = (1 + x^2 * (1+x)^3),
F(3) = (1 + x^4 * (1+x)^6 * F(2)^3),
F(4) = (1 + x^8 * (1+x)^12 * F(2)^6 * F(3)^3),
F(5) = (1 + x^16 * (1+x)^24 * F(2)^12 * F(3)^6 * F(4)^3),
...
F(n+1) = 1 + (F(n) - 1)^2 * F(n)^3
...
Then the g.f. A(x) equals the infinite product:
A(x) = x * F(1)^3 * F(2)^3 * F(3)^3 * ... * F(n)^3 * ...
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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)^3 ) - 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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