OFFSET
1,4
COMMENTS
All terms are odd.
Limit a(n)/(n-1)! = 0.537518285445620493117044794174835470291338509726127442574604...
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..400
FORMULA
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x*A(x) - x^3*A'(x)) = x^2.
(2) A(x) = x^2*A'(x) + B(x^2)/x, where B(A(x)) = x.
(3) a(2*n) = (2*n-1)*a(2*n-1) for n >= 0.
(4) a(2*n+1) = (2*n)*a(2*n) + [x^(n+1)] B(x) for n >= 0, where B(A(x)) = x.
EXAMPLE
G.f.: A(x) = x + x^2 + x^3 + 3*x^4 + 13*x^5 + 65*x^6 + 387*x^7 + 2709*x^8 + 21673*x^9 + 195057*x^10 + 1950547*x^11 + 21456017*x^12 + ...
where A(x*A(x) - x^3*A'(x)) = x^2.
RELATED SERIES.
The derivative of A(x) wrt x begins
A'(x) = 1 + 2*x + 3*x^2 + 12*x^3 + 65*x^4 + 390*x^5 + 2709*x^6 + 21672*x^7 + 195057*x^8 + 1950570*x^9 + 21456017*x^10 + ...
where [x^(2*n)] A'(x) = [x^(2*n+2)] A(x) for n >= 0.
Let B(x) satisfy B(A(x)) = x, then
B(x) = x - x^2 + x^3 - 3*x^4 + x^5 - 23*x^6 - 81*x^7 - 747*x^8 - 5837*x^9 - 55467*x^10 - 568279*x^11 - 6414523*x^12 + ...
where B(x^2) = x*A(x) - x^3*A'(x).
PROG
(PARI) {a(n) = my(V=[0, 1], A=x); for(i=1, n, V = concat(V, 0); A = Ser(V);
V[#V] = polcoef(x^2 - subst(A, x, x*A - x^3*A' ), #V) ); polcoef(A, n)}
for(n=1, 40, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 04 2024
STATUS
approved