OFFSET
0,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..500
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
(1) A(x) = Sum_{n=-oo..+oo} (-x*A(x))^n * (1 - (-x*A(x))^(n-1))^n.
(2) A(x) = Sum_{n=-oo..+oo} (-x*A(x))^n * (1 - (-x*A(x))^(n-1))^(n+1).
(3) A(x) = Sum_{n=-oo..+oo} (x*A(x))^(2*n+1) * (1 - (-x*A(x))^n)^n.
(4) A(x) = Sum_{n=-oo..+oo} (x*A(x))^(n^2) / (1 - (-x*A(x))^(n+1))^(n+1).
(5) x*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * (x*A(x))^(n*(n-1)) / (1 - (-x*A(x))^(n+1))^(n-1).
(6) -1/x = Sum_{n=-oo..+oo} (-x*A(x))^n * (1 - (-x*A(x))^n)^(n+1).
(7) -1/x = Sum_{n=-oo..+oo} (-x*A(x))^n * (1 - (-x*A(x))^n)^(n+2).
(8) 1/x = Sum_{n=-oo..+oo} (x*A(x))^(2*n) * (1 - (-x*A(x))^n)^n.
(9) 0 = Sum_{n=-oo..+oo} (-x*A(x))^n * (1 - (-x*A(x))^n)^n.
(10) 0 = Sum_{n=-oo..+oo} (x*A(x))^(2*n) * (1 - (-x*A(x))^n)^(n+1).
(11) A(-x/G(x)) = G(x) where G(x) = Sum_{n=-oo..+oo} (x - x^n)^n is the g.f. of A290003.
(12) A(x) = (-1/x) * Series_Reversion( -x / Sum_{n=-oo..+oo} (x - x^n)^n ).
a(n) ~ c * d^n / n^(3/2), where d = 4.4911010651615255101195452998052055698... and c = 0.53507007927413038001531299966030791... - Vaclav Kotesovec, Mar 14 2023
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 20*x^4 + 69*x^5 + 245*x^6 + 896*x^7 + 3362*x^8 + 12869*x^9 + 50024*x^10 + 196896*x^11 + 783205*x^12 + ...
SPECIFIC VALUES.
A(x) = 2 at x = 0.22210374835192555734961892166866769267669905135315...
A(1/5) = 1.45174689360673617561694352881716190508117725206270...
A(1/6) = 1.28852385900727494844427701605174847197781970881818...
PROG
(PARI) {a(n) = my(A=1);
A = (-1/x)*serreverse(-x/sum(m=-n-1, n+1, (x - x^m +x*O(x^(n+1)))^m )); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=1); for(i=1, n,
A = sum(m=-n, n, (-x*A)^m * (1 - (-x*A)^(m-1) +x*O(x^n))^m)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 17 2023
STATUS
approved