OFFSET
1,2
COMMENTS
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..520
FORMULA
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1.a) A(x)^2 = A( x^2 + 4*A(x)^3 ).
(1.b) A(x)^4 = A( x^4 + 8*x^2*A(x)^3 + 20*A(x)^6 ).
(1.c) A(x)^8 = A( x^8 + 16*x^6*A(x)^3 + 104*x^4*A(x)^6 + 320*x^2*A(x)^9 + 404*A(x)^12 ).
(1.d) A(x)^16 = A( x^16 + 32*x^14*A(x)^3 + 464*x^12*A(x)^6 + 3968*x^10*A(x)^9 + 21864*x^8*A(x)^12 + 79488*x^6*A(x)^15 + 186432*x^4*A(x)^18 + 258560*x^2*A(x)^21 + 163220*A(x)^24 ).
(2.a) A(x - x*B(x)) = x where B(x) = 2*x + (B(x)^2 + B(x^2))/2 is the g.f. of A083563, which enumerates expressions in at most 2 variables.
(2.b) A(x) = x + Sum_{n>=0} d^n/dx^n (x*B(x))^(n+1)/(n+1)! where B(x) is the g.f. of A083563.
(2.c) A(x) = x * exp( Sum_{n>=0} d^n/dx^n x^n*B(x)^(n+1)/(n+1)! ) where B(x) is the g.f. of A083563.
(3.a) A(x) = F( x^2/A(x) ) where F(x)^2 = F( (x + 4*F(x)^2)^2 ) and F(x) is the g.f. of A386667.
(3.b) A( sqrt(x*F(x)) ) = F(x) where F(x) is the g.f. of A386667.
(3.c) A( x*F(x) + 4*F(x)^3 ) = F(x)^2 where F(x) is the g.f. of A386667.
EXAMPLE
G.f. A(x) = x + 2*x^2 + 11*x^3 + 76*x^4 + 593*x^5 + 4968*x^6 + 43659*x^7 + 397044*x^8 + 3705095*x^9 + 35277250*x^10 + 341347592*x^11 + 3346919876*x^12 + ...
where A(x)^2 = A( x^2 + 4*A(x)^3 ).
RELATED SERIES.
A(x)^2 = x^2 + 4*x^3 + 26*x^4 + 196*x^5 + 1611*x^6 + 13980*x^7 + 126012*x^8 + 1168156*x^9 + 11065649*x^10 + 106638064*x^11 + 1042127560*x^12 + ...
A(x)^3 = x^3 + 6*x^4 + 45*x^5 + 368*x^6 + 3186*x^7 + 28674*x^8 + 265538*x^9 + 2513472*x^10 + 24208029*x^11 + 236467516*x^12 + ...
The square root of x*A(x) appears to be an integer series:
sqrt(x*A(x)) = x + x^2 + 5*x^3 + 33*x^4 + 251*x^5 + 2068*x^6 + 17962*x^7 + 161937*x^8 + 1501056*x^9 + 14216070*x^10 + 136961751*x^11 + 1338091386*x^12 + ...
The series reversion of A(x) = x*(1 - B(x)) where B(x) = 2*x + (B(x)^2 + B(x^2))/2 is the g.f. of A083563:
B(x) = 2*x + 3*x^2 + 6*x^3 + 18*x^4 + 54*x^5 + 183*x^6 + 636*x^7 + 2316*x^8 + 8610*x^9 + 32763*x^10 + ... + A083563(n)*x^n + ...
Let F(x) satisfy F(x) = A( sqrt(x*F(x)) ), then
F(x) = x + 4*x^2 + 34*x^3 + 360*x^4 + 4271*x^5 + 54292*x^6 + 723030*x^7 + 9957288*x^8 + 140645083*x^9 + ... + A386667(n)*x^n + ...
where F(x)^2 = F( (x + 4*F(x)^2)^2 ).
SPECIFIC VALUES.
A(t) = 1/7 at t = 0.08880743852136549481708047043610490271007905442085213... where 1/49 = A( t^2 + 4/7^3 ).
A(t) = 1/8 at t = 0.08551256888153515385269624564322311398904048542773525... where 1/64 = A( t^2 + 1/128 ).
A(t) = 1/10 at t = 0.0761376263361552038603962313195463869965398686252210... where 1/100 = A( t^2 + 1/250 ).
A(t) = 1/12 at t = 0.0673213195608991751861324471604892527235665458784846... where 1/144 = A( t^2 + 1/432 ).
A(1/12) = 0.1177955673566540556266475808076641068445182181357392... where A(1/12)^2 = A( 1/144 + 4*A(1/12)^3 ).
A(1/14) = 0.0906630585422134136382590608695361754169902539743029... where A(1/14)^2 = A( 1/196 + 4*A(1/14)^3 ).
A(1/16) = 0.0754118068060563404285214280710447911081276500103364... where A(1/16)^2 = A( 1/256 + 4*A(1/16)^3 ).
PROG
(PARI) {a(n) = my(A=x+x^2, X=x+x*O(x^n)); for(i=1, n, A = subst(A, x, X^2 + 4*A^3)^(1/2) ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 07 2025
STATUS
approved
