OFFSET
0,4
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..300
FORMULA
G.f. satisfies: x*A'(x) = A(x)*(1+x - A(x))/(A(x) - 1).
G.f.: A(x) = 1/G(-x) where G(x) is the g.f. of A088715.
G.f. satisfies: A(x/F(x)) = F(x) where F(x) is the g.f. of A158883.
G.f. satisfies: A(x*H(-x)) = H(-x) where H(x) is the g.f. of A088716.
G.f. satisfies: [x^n] 1/A(-x)^(n+2) = [x^(n+1)] 1/A(-x)^(n+2)/(n+2) = A088716(n+1).
a(n) ~ -(-1)^n * c * n! * n^2, where c = A238223 / exp(1) = 0.080179614624692622... - Vaclav Kotesovec, Nov 21 2017
a(0) = 1; a(n) = a(n-1) - (n+2)/2 * Sum_{k=1..n-1} a(k) * a(n-k). - Seiichi Manyama, Apr 04 2026
EXAMPLE
G.f.: A(x) = 1 + x - x^2 + 4*x^3 - 23*x^4 + 166*x^5 - 1410*x^6 +...
d/dx x*A(x) = 1 + 2*x - 3*x^2 + 16*x^3 - 115*x^4 + 996*x^5 - 9870*x^6 +...
d/dx log(A(x)) = 1 - 3*x + 16*x^2 - 115*x^3 + 996*x^4 - 9870*x^5 +...
Coefficients in powers A(x)^-n begin:
A(x)^-1: (1), -1, 2, -7, 36, -240, 1926, -17815, 184916, ...
A(x)^-2: (1),(-2), 5, -18, 90, -580, 4525, -40946, 417822, ...
A(x)^-3: 1, (-3), (9), -34, 168, -1053, 7997, -70776, 709614, ...
A(x)^-4: 1, -4, (14),(-56), 277, -1700, 12594, -109032, 1073658, ...
A(x)^-5: 1, -5, 20, (-85),(425), -2571, 18630, -157860, 1526330, ...
A(x)^-6: 1, -6, 27, -122, (621),(-3726), 26492, -219912, 2087658, ...
A(x)^-7: 1, -7, 35, -168, 875, (-5236),(36652), -298446, 2782080, ...
A(x)^-8: 1, -8, 44, -224, 1198, -7184, (49680),(-397440),3639333, ...
...
and A(x)^-1 (first row) is the g.f. of signed A088715.
PROG
(PARI) {a(n)=local(A=[1, 1]); for(i=2, n, A=concat(A, 0); A[ #A]=(Vec(Ser(A)^(#A-1))-Vec(Ser(A)^(#A)))[ #A]); Vec(Ser(A)^(n+1)/(n+1))[n+1]}
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Apr 30 2009
STATUS
approved