OFFSET
1,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..300
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)^3).
(2) A(x) = x + A(x)^3 * A'(A(x))^3.
(3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(3*n-1) * A'(x)^(3*n) / n! ).
(4) A'(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A'(x)^(3*n) / n! is the g.f. of A361536.
(5) a(n) = A361536(n-1)/(2*n-1) for n >= 1.
EXAMPLE
G.f.: A(x) = x + x^3 + 12*x^5 + 291*x^7 + 10243*x^9 + 460632*x^11 + 24830853*x^13 + 1546531419*x^15 + 108716955930*x^17 + ... + a(n)*x^(2*n-1) + ...
By definition, A(x - x^3*A'(x)^3) = x, where
A'(x) = 1 + 3*x^2 + 60*x^4 + 2037*x^6 + 92187*x^8 + 5066952*x^10 + 322801089*x^12 + 23197971285*x^14 + ... + A361536(n)*x^(2*n) + ...
Also,
A'(x) = 1 + (d/dx x^3*A'(x)^3) + (d^2/dx^2 x^6*A'(x)^6)/2! + (d^3/dx^3 x^9*A'(x)^9)/3! + (d^4/dx^4 x^12*A'(x)^12)/4! + (d^5/dx^5 x^15*A'(x)^15)/5! + ... + (d^n/dx^n x^(3*n)*A'(x)^(3*n))/n! + ...
Further,
A(x) = x * exp( x^2*A'(x)^3 + (d/dx x^5*A'(x)^6)/2! + (d^2/dx^2 x^8*A'(x)^9)/3! + (d^3/dx^3 x^11*A'(x)^12)/4! + (d^4/dx^4 x^14*A'(x)^15)/5! + ... + (d^(n-1)/dx^(n-1) x^(3*n-1)*A'(x)^(3*n))/n! + ... ).
PROG
(PARI) {a(n) = my(A=x+x^3); for(i=1, n, A = serreverse(x - x^3*A'^3 +x*O(x^(2*n)))); polcoeff(A, 2*n-1)}
for(n=1, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 17 2023
STATUS
approved