OFFSET
1,2
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, along with B(x) where B(A(x)) = x, satisfies the following formulas.
(1) B(x) = B(B(x)^2) / (x - 5*x^2).
(2) B(x^2) = x*A(x) - 5*x*A(x)^2.
(3) A(x) - 5*A(x)^2 is an odd function.
(4) A(x) = (1 - sqrt(1 - 20*B(x^2)/x)) / 10.
(5) A(x) = (1/5)*C( 5*B(x^2)/x ), where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers A000108.
The radius of convergence r satisfies r^2 = A(r/20), A(r) = 1/10, and A(-r) = -(sqrt(2) - 1)/10 where r = 0.050648013557126794001613224411704207390680989556692787...
EXAMPLE
G.f.: A(x) = x + 5*x^2 + 45*x^3 + 575*x^4 + 8005*x^5 + 118925*x^6 + 1848175*x^7 + 29683375*x^8 + 488787945*x^9 + 8207945825*x^10 + ...
such that A( x*A(x) - 5*x*A(x)^2 ) = x^2
where
A(x)^2 = x^2 + 10*x^3 + 115*x^4 + 1600*x^5 + 23785*x^6 + 369650*x^7 + 5936675*x^8 + 97757500*x^9 + 1641589165*x^10 + ...
and
A(x) - 5*A(x)^2 = x - 5*x^3 + 5*x^5 - 75*x^7 + 445*x^9 - 2375*x^11 + 18875*x^13 - 135625*x^15 + 971555*x^17 - 7626225*x^19 + ...
which is an odd function.
Compare with B(x), the series reversion of A(x), A(B(x)) = x:
B(x) = x - 5*x^2 + 5*x^3 - 75*x^4 + 445*x^5 - 2375*x^6 + 18875*x^7 - 135625*x^8 + 971555*x^9 - 7626225*x^10 + ...
which satisfies: B(B(x)^2) = (x - 5*x^2)*B(x).
SPECIFIC VALUES.
A(1/20) = 0.0888331272418844744627229655682642865491084541976232...
A(1/21) = 0.0758352831248204153051711824829869843447396211754731...
A(1/22) = 0.0683327114842221656011473194179884705760694623166237...
where 1/22^2 = A( (1/22)*(A(1/22) - 5*A(1/22)^2) ).
PROG
(PARI) {a(n, k=5) = my(A=x); for(i=1, #binary(n), A = (1 - sqrt(1 - 4*k*subst(serreverse(A +x*O(x^n)), x, x^2)/x) )/(2*k) ); polcoeff(GF=A, n)}
{upto(n) = a(n, 5); Vec(GF)}
upto(25)
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 03 2026
STATUS
approved