OFFSET
1,3
COMMENTS
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).
More generally, for prime p there exists an integer series G(x) that satisfies: G(x) = G( x^p + p*x*G(x)^p )^(1/p) with G(0)=0, G'(0)=1 (conjecture).
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..500
EXAMPLE
G.f.: A(x) = x + x^2 + 3*x^3 + 11*x^4 + 44*x^5 + 185*x^6 + 803*x^7 + 3564*x^8 + 16082*x^9 + 73502*x^10 + 339391*x^11 + 1580318*x^12 + ...
where A(x)^5 = A( x^5 + 5*x*A(x)^5 ).
RELATED SERIES.
A(x)^5 = x^5 + 5*x^6 + 25*x^7 + 125*x^8 + 625*x^9 + 3126*x^10 + 15640*x^11 + 78275*x^12 + 391875*x^13 + 1962500*x^14 + 9831253*x^15 + 49265695*x^16 + ...
Let R(x) be the series reversion of g.f. A(x), R(A(x)) = x, then R(x) begins
R(x) = x - x^2 - x^3 - x^4 + 4*x^6 + 15*x^7 + 36*x^8 + 55*x^9 - 359*x^11 - 1520*x^12 - 4028*x^13 - 6667*x^14 + 49062*x^16 + 217645*x^17 + ...
The 4 quintisections of R(x) = Q1(x) + Q2(x) + Q3(x) + Q4(x) (with the fifth being zero) are as follows
Q1(x) = x + 4*x^6 - 359*x^11 + 49062*x^16 - 8013396*x^21 + 1442958557*x^26 - 276352605126*x^31 + 55224710824185*x^36 - 11384289478228711*x^41 + ...
Q2(x) = -x^2 + 15*x^7 - 1520*x^12 + 217645*x^17 - 36405005*x^22 + 6650838668*x^27 - 1286179025729*x^32 + 258819346825534*x^37 + ...
Q3(x) = -x^3 + 36*x^8 - 4028*x^13 + 600254*x^18 - 102567034*x^23 + 18988120493*x^28 - 3705388523045*x^33 + 750546817970646*x^38 + ...
Q4(x) = -x^4 + 55*x^9 - 6667*x^14 + 1028514*x^19 - 179152944*x^24 + 33573744984*x^29 - 6607215559460*x^34 + 1346634048063165*x^39 + ...
where Q1*Q4 = -Q2*Q3 where
Q2*Q3 = x^5 - 51*x^10 + 6088*x^15 - 933039*x^20 + 161933629*x^25 - 30277104991*x^30 + 5949003081867*x^35 - 1211076410858363*x^40 + ...
PROG
(PARI) {a(n) = my(A=x+x^2, X=x+x*O(x^n)); for(i=1, n, A = subst(A, x, x^5 + 5*X*A^5)^(1/5) ); polcoeff(A, n)}
for(n=1, 40, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 16 2016
STATUS
approved