OFFSET
1,3
COMMENTS
a(n) = A360973(n-1)/(2*n-1) for n >= 1.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..200
FORMULA
G.f. A(x) = Sum_{n>=1} a(n)*x^(2*n-1) may be defined by the following.
(1) A(x) = Series_Reversion(x - x^3*A'(x)).
(2) A(x) = x + A(x)^3 * A'(A(x)).
(3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(3*n-1) * A'(x)^n / n! ).
(4) A'(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A'(x)^n / n!.
EXAMPLE
G.f.: A(x) = x + x^3 + 6*x^5 + 66*x^7 + 1027*x^9 + 20274*x^11 + 479403*x^13 + 13118703*x^15 + 406181493*x^17 + ... + a(n)*x^(2*n-1) + ...
By definition, A(x - x^3*A'(x)) = x, where
A'(x) = 1 + 3*x^2 + 30*x^4 + 462*x^6 + 9243*x^8 + 223014*x^10 + 6232239*x^12 + 196780545*x^14 + ... + A360973(n)*x^(2*n) + ...
Also,
A'(x) = 1 + (d/dx x^3*A'(x)) + (d^2/dx^2 x^6*A'(x)^2)/2! + (d^3/dx^3 x^9*A'(x)^3)/3! + (d^4/dx^4 x^12*A'(x)^4)/4! + (d^5/dx^5 x^15*A'(x)^5)/5! + (d^6/dx^6 x^18*A'(x)^6)/6! + ... + (d^n/dx^n x^(3*n)*A'(x)^n)/n! + ...
Further,
A(x) = x * exp( x^2*A'(x) + (d/dx x^5*A'(x)^2)/2! + (d^2/dx^2 x^8*A'(x)^3)/3! + (d^3/dx^3 x^11*A'(x)^4)/4! + (d^4/dx^4 x^14*A'(x)^5)/5! + (d^5/dx^5 x^17*A'(x)^6)/6! + ... + (d^(n-1)/dx^(n-1) x^(3*n-1)*A'(x)^n)/n! + ... ).
PROG
(PARI) {a(n) = my(A=x+x^2); for(i=1, n, A=serreverse(x - x^3*A'+x*O(x^(2*n)))); polcoeff(A, 2*n-1)}
for(n=1, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 27 2023
STATUS
approved