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).
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..400
EXAMPLE
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).
PROG
(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), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 26 2024
STATUS
approved