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A369556
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Expansion of g.f. A(x) satisfying A(x) = A( x^2*(1+x)^6 ) / x.
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7
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1, 6, 21, 92, 432, 1704, 6276, 24096, 95628, 383848, 1560030, 6395256, 26220902, 106855404, 430894908, 1716023728, 6758075160, 26395681392, 102540292584, 397219456608, 1538055130419, 5963874635622, 23183457031431, 90385003122912, 353392849642574, 1385262648293892, 5441942144992950
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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)^6 = 1 where r = 0.2554228710768465432050...
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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)^6 ) / x.
(2) R(x*A(x)) = x^2*(1+x)^6, where R(A(x)) = x.
(3) A(x) = x * Product_{n>=1} F(n)^6, where F(1) = 1+x, and F(n+1) = 1 + (F(n) - 1)^2 * F(n)^6 for n >= 1.
(4) A(x) = B(x)^6/x^5 where B(x) is the g.f. of A369549.
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EXAMPLE
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G.f.: A(x) = 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 + 383848*x^10 + 1560030*x^11 + 6395256*x^12 + ...
RELATED SERIES.
(x^5*A(x))^(1/6) = 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 + ... + A369549(n)*x^n + ...
Let R(x) be the series reversion of A(x),
R(x) = x - 6*x^2 + 51*x^3 - 542*x^4 + 6471*x^5 - 82428*x^6 + 1095952*x^7 - 15036582*x^8 + 211325931*x^9 - 3026813166*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)^6.
GENERATING METHOD.
Define F(n), a polynomial in x of order 8^(n-1), by the following recurrence:
F(1) = (1 + x),
F(2) = (1 + x^2 * (1+x)^6),
F(3) = (1 + x^4 * (1+x)^12 * F(2)^6),
F(4) = (1 + x^8 * (1+x)^24 * F(2)^12 * F(3)^6),
F(5) = (1 + x^16 * (1+x)^48 * F(2)^24 * F(3)^12 * F(4)^6),
...
F(n+1) = 1 + (F(n) - 1)^2 * F(n)^6
...
Then the g.f. A(x) equals the infinite product:
A(x) = x * F(1)^6 * F(2)^6 * F(3)^6 * ... * F(n)^6 * ...
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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*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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