|
EXAMPLE
|
G.f.: A(x) = 1 + x - 2*x^2 + 9*x^3 - 56*x^4 + 425*x^5 - 3726*x^6 + ...
(d/dx) (x/A(x)) = 1 - 2*x + 9*x^2 - 56*x^3 + 425*x^4 - 3726*x^5 + ...
1/A(x) = 1 - x + 3*x^2 - 14*x^3 + 85*x^4 + ... + (-1)^n*A088716(n)*x^n + ...
where a(n) = (-1)^(n-1)*n*A088716(n-1) for n >= 1.
...
Coefficients of powers of g.f. A(x) begin:
A^1: 1,1,-2,9,-56,425,-3726,36652,-397440,4695489,...;
A^2: 1,2,(-3),14,-90,702,-6297,63144,-695886,8334822,...;
A^3: 1,3,(-3),(16),-108,870,-7997,81774,-915798,11116902,...;
A^4: 1,4,-2,(16),(-115),960,-9050,94368,-1073658,13204560,...;
A^5: 1,5,0,15,(-115),(996),-9630,102365,-1182690,14730890,...;
A^6: 1,6,3,14,-111,(996),(-9870),106890,-1253466,15804548,...;
A^7: 1,7,7,14,-105,973,(-9870),(108816),-1294412,16514162,...;
A^8: 1,8,12,16,-98,936,-9704,(108816),(-1312227),16931984,...;
A^9: 1,9,18,21,-90,891,-9426,107406,(-1312227),(17116900),...;
A^10:1,10,25,30,-80,842,-9075,104980,-1298625,(17116900),...; ...
where coefficients [x^n] A(x)^(n+1) and [x^n] A(x)^n are
enclosed in parenthesis and equal (n+1)*A158884(n) for n > 1:
[ -3,16,-115,996,-9870,108816,-1312227,17116900,...];
[1,1,-1,4,-23,166,-1410,13602,-145803,1711690,-21785618,...]
and also to the logarithmic derivative of A158884:
[1,-3,16,-115,996,-9870,108816,-1312227,17116900,...].
|