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A370545
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Expansion of g.f. A(x) satisfying A(x) = A( x^5 + 5*A(x)^6 )^(1/5).
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2
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1, 1, 4, 21, 125, 801, 5388, 37518, 268109, 1955000, 14487754, 108794169, 826054062, 6331064385, 48914088750, 380555960864, 2978892961194, 23444095375593, 185394136871818, 1472396312841250, 11739089289817538, 93921736129064325, 753845680317416682, 6068255413854119432
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OFFSET
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1,3
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COMMENTS
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Compare the g.f. to the following identities:
(1) C(x) = C( x^2 + 2*x*C(x)^2 )^(1/2),
(2) C(x) = C( x^3 + 3*x*C(x)^3 )^(1/3),
where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).
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LINKS
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EXAMPLE
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G.f.: A(x) = x + x^2 + 4*x^3 + 21*x^4 + 125*x^5 + 801*x^6 + 5388*x^7 + 37518*x^8 + 268109*x^9 + 1955000*x^10 + 14487754*x^11 + 108794169*x^12 + ...
where A(x)^5 = A( x^5 + 5*A(x)^6 ).
RELATED SERIES.
A(x)^5 = x^5 + 5*x^6 + 30*x^7 + 195*x^8 + 1330*x^9 + 9376*x^10 + 67720*x^11 + ...
A(x)^6 = x^6 + 6*x^7 + 39*x^8 + 266*x^9 + 1875*x^10 + 13542*x^11 + 99654*x^12 + ...
Let B(x) be the series reversion of A(x), A(B(x)) = x, which begins
B(x) = x - x^2 - 2*x^3 - 6*x^4 - 21*x^5 - 80*x^6 - 320*x^7 - 1326*x^8 - 5637*x^9 - 24434*x^10 - ... + (-1)^(n-1)*A352703(n-1)*x^n + ...
then B(x)^5 + 5*x^6 = B(x^5).
Let C(x) = x^2/B(x) = x + x^2 + 3*x^3 + 11*x^4 + 44*x^5 + 185*x^6 + 802*x^7 + 3553*x^8 + 15994*x^9 + 72886*x^10 + ... + A091200(n-1)*x^n + ...
where A(x^2/C(x)) = x and C(A(x)) = A(x)^2/x,
then C(x)^5 = C(x^5)/(1 - 5*C(x^5)/x^4).
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PROG
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(PARI) {a(n) = my(A=x+x^2); for(m=1, n, A=truncate(A); A = subst(A, x, x^5 + 5*A^6 +x^5*O(x^m))^(1/5) ); polcoeff(A, n)}
for(n=1, 40, 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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