OFFSET
1,3
COMMENTS
Compare to: if C(x) = x + C(x)^2 is the g.f. of the Catalan numbers (A000108), then
(C.1) C(x)^3 = C( x^3 + 3*x^2*C(x)^2 + 3*x*C(x)^4 ),
(C.2) C(x)^3 = C( x^3 + 3*x*C(x)^3 ).
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..500
FORMULA
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x)^3 = A( x^3 + 3*x*A(x)^3 + 3*A(x)^5 ).
(2) B(x^3) = B(x)^3 + 3*x^3*B(x) + 3*x^5 where B(A(x)) = x.
EXAMPLE
G.f.: A(x) = x + x^2 + 3*x^3 + 11*x^4 + 45*x^5 + 197*x^6 + 903*x^7 + 4278*x^8 + 20780*x^9 + 102932*x^10 + 517958*x^11 + 2640351*x^12 + ...
where A(x)^3 = A( x^3 + 3*x*A(x)^3 + 3*A(x)^5 ).
RELATED SERIES.
A(x)^3 = x^3 + 3*x^4 + 12*x^5 + 52*x^6 + 237*x^7 + 1119*x^8 + 5424*x^9 + 26829*x^10 + 134868*x^11 + 686993*x^12 + ...
A(x)^5 = x^5 + 5*x^6 + 25*x^7 + 125*x^8 + 630*x^9 + 3206*x^10 + 16470*x^11 + 85345*x^12 + ...
x^3 + 3*x*A(x)^3 + 3*A(x)^5 = x^3 + 3*x^4 + 12*x^5 + 51*x^6 + 231*x^7 + 1086*x^8 + 5247*x^9 + 25890*x^10 + 129897*x^11 + 660639*x^12 + ...
Let B(x) be the series reversion of g.f. A(x), so that B(A(x)) = x, then
B(x) = x - x^2 - x^3 - x^4 - x^5 - x^6 - x^7 + 2*x^9 + 6*x^10 + 14*x^11 + 27*x^12 + 46*x^13 + 70*x^14 + 89*x^15 + 77*x^16 - 24*x^17 - 331*x^18 - 1048*x^19 + ...
where B(x^3) = B(x)^3 + 3*x^3*B(x) + 3*x^5.
SPECIFIC VALUES.
A(t) = 1/4 at t = 0.16670634316087722742289913727394381851520339572385...
A(t) = 1/5 at t = 0.15000528806976999341591230789366585995826596594852...
A(t) = 1/6 at t = 0.13333440108660223078452472482440873571302766056370...
A(1/6) = 0.24982295070218650636257016421484515902901372739970...
A(1/7) = 0.18469374868046130312396796140380302078509253618624...
A(1/8) = 0.15240213894190631145603541851084182452570637444970...
PROG
(PARI) {a(n) = my(A=[0, 1], Ax=x); for(i=1, n, A = concat(A, 0); Ax=Ser(A);
A[#A] = (1/3)*polcoeff( subst(Ax, x, x^3 + 3*x*Ax^3 + 3*Ax^5) - Ax^3, #A+1) ); A[n+1]}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 20 2024
STATUS
approved