OFFSET
1,3
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, along with its series reversion R(x), satisfy the following formulas.
(1) A( -x*A(x - x^2) ) = -x^2.
(2) A(x - x^2) = R(-x^2)/(-x).
(3) (R(x) - R(-x))^2 + 2*(R(x) + R(-x)) = 0.
(4) R(x) = R(-x) - 1 + sqrt(1 - 4*R(-x)).
(5) A(x) = -A( x - 1 + sqrt(1 - 4*x) ).
(6) A(x) = -A(x - 2*C(x)) where C(x) = -C(x - 2*C(x)) is a g.f. of the Catalan numbers (A000108).
(7) A( -A(x)*C(x) ) = -C(x)^2 where C(x) = (1 - sqrt(1 - 4*x))/2 is a g.f. of the Catalan numbers (A000108).
EXAMPLE
G.f.: A(x) = x + x^2 + 3*x^3 + 8*x^4 + 22*x^5 + 65*x^6 + 200*x^7 + 637*x^8 + 2090*x^9 + 7021*x^10 + 24041*x^11 + 83611*x^12 + ...
RELATED SERIES.
Let R(x) be the series reversion of g.f. A(x), R(A(x)) = x, then
R(x) = x - x^2 - x^3 + 2*x^4 + 4*x^5 - 9*x^6 - 18*x^7 + 44*x^8 + 91*x^9 - 234*x^10 - 496*x^11 + 1318*x^12 + ...
where A(x - x^2) = R(-x^2)/(-x).
Also, the bisections B1 and B2 of R(x) are
B1 = (R(x) - R(-x))/2 = x - x^3 + 4*x^5 - 18*x^7 + 91*x^9 - 496*x^11 + 2839*x^13 - 16836*x^15 + 102545*x^17 - 637733*x^19 + ...
B2 = (R(x) + R(-x))/2 = -x^2 + 2*x^4 - 9*x^6 + 44*x^8 - 234*x^10 + 1318*x^12 - 7722*x^14 + 46594*x^16 - 287611*x^18 + 1807720*x^20 + ...
and satisfy B1^2 + B2 = 0 and A(-x*B1) = -B1^2.
SPECIFIC VALUES.
A( -A(2/9) / 3 ) = -1/9 where
A(2/9) = 0.3655811677545134614272600644874552972994602150418984...
A( -A(3/16) / 4 ) = -1/16 where
A(3/16) = 0.2645434685642398513217156896362957133168212272114320...
A( -A(4/25) / 5 ) = -1/25 where
A(4/25) = 0.2076566162630115730635446744577181791494166261819659...
A( -A(5/36) / 6 ) = -1/36 where
A(5/36) = 0.1711609712404346976409014231532840797963445277760447...
PROG
(PARI) {a(n) = my(A=[0, 1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff( x^2 + subst(Ser(A), x, -x*subst(Ser(A), x, x - x^2) ), #A)); A[n+1]}
for(n=1, 35, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 05 2024
STATUS
approved