OFFSET
1,5
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..969
FORMULA
G.f. A(x) = Sum_{n>=1} a(n) * x^n satisfies the following formulas.
(1) A(x) = A( x^3 + 3*x^2*A(x)^2 )^(1/3).
(2) B(x^3) = B(x)^3 + 3*x^2*B(x)^2, where A(B(x)) = x.
a(n) ~ c * d^n / n^(3/2), where d = 2.653503750287... and c = 0.193303... - Vaclav Kotesovec, Mar 14 2024
EXAMPLE
G.f.: A(x) = x + x^2 + x^3 + x^4 + 2*x^5 + 6*x^6 + 15*x^7 + 30*x^8 + 55*x^9 + 113*x^10 + 274*x^11 + 683*x^12 + 1596*x^13 + 3547*x^14 + 7990*x^15 + ...
where A(x)^3 = A( x^3 + 3*x^2*A(x)^2 ).
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 7*x^6 + 18*x^7 + 47*x^8 + 106*x^9 + 216*x^10 + 450*x^11 + 1040*x^12 + ...
A(x)^3 = x^3 + 3*x^4 + 6*x^5 + 10*x^6 + 18*x^7 + 42*x^8 + 109*x^9 + 264*x^10 + 585*x^11 + 1270*x^12 + ...
Let B(x) denote the series reversion of A(x), A(B(x)) = x,
B(x) = x - x^2 + x^3 - x^4 + x^6 - x^7 + 2*x^9 - 3*x^10 + 6*x^12 - 9*x^13 + 20*x^15 - 30*x^16 + 71*x^18 - 110*x^19 + 267*x^21 - 419*x^22 + 1041*x^24 - 1648*x^25 + 4168*x^27 - 6652*x^28 + 17047*x^30 + ...
then B(x^3) = B(x)^3 + 3*x^2*B(x)^2, where
B(x)^2 = x^2 - 2*x^3 + 3*x^4 - 4*x^5 + 3*x^6 - 3*x^8 + 4*x^9 - 8*x^11 + 11*x^12 - 23*x^14 + 34*x^15 + ...
B(x)^3 = x^3 - 3*x^4 + 6*x^5 - 10*x^6 + 12*x^7 - 9*x^8 + x^9 + 9*x^10 - 12*x^11 - x^12 + 24*x^13 - 33*x^14 + 69*x^16 - 102*x^17 + ...
Further, the trisections of B(x) = C1(x) + C2(x) + C3(x) are
C1(x) = x^4/C3(x) = x - x^4 - x^7 - 3*x^10 - 9*x^13 - 30*x^16 - 110*x^19 - ...
C2(x) = -x^2, and
C3(x) = x^3 + x^6 + 2*x^9 + 6*x^12 + 20*x^15 + 71*x^18 + 267*x^21 + 1041*x^24 + 4168*x^27 + 17047*x^30 + 70902*x^33 + ... + A370446(n)*x^(3*n) + ...
Compare these series to the series trisections involved in series reversion of A264228.
SPECIFIC VALUES.
A(1/3) = 0.5339969110985873619406256103732700685272...
A(1/4) = 0.3373018860609501862067597141160425025580...
A(1/5) = 0.2509433336474255853462277222741392614966...
A(1/6) = 0.2003115176013404351183299069966738623357...
A(1/8) = 0.1429156905534693639298206599148805278651...
A(1/3)^3 = A(1/27 + 3*A(1/3)^2/9) = A(0.132087937391...) = 0.152270661558...
A(1/4)^3 = A(1/64 + 3*A(1/4)^2/16) = A(0.036957355438...) = 0.038375699859...
A(1/5)^3 = A(1/125 + 3*A(1/5)^2/25) = A(0.015556706804...) = 0.250943333647...
PROG
(PARI) {a(n) = my(A=[1], G); for(i=1, n, G = x*Ser(A); A = Vec((subst(G, x, x^3 + 3*x^2*G^2) + x^4*O(x^#A))^(1/3)); ); A[n+1]}
for(n=0, 40, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 09 2024
STATUS
approved