OFFSET
0,2
COMMENTS
The g.f. of A167003, G(x), satisfies:
Sum_{n>=0} log(G(3^n*x))^n/n! = 1 + Sum_{n>=0} 3^(3^n-n)*x^(3^n).
FORMULA
a(n) = [x^n] G(x)^(3^(n+1)) for n>=0 where G(x) = g.f. of A167003.
EXAMPLE
G.f.: A(x) = 1 + 9*x + 243*x^2 + 59076*x^3 + 111615732*x^4 +...
Let G(x) equal the g.f. of A167003:
G(x) = 1 + x - 4*x^2 - 4*x^3 - 8220*x^4 - 16910960*x^5 - 220513689396*x^6 +...
then the g.f. A(x) of this sequence equals the series:
A(x) = 1 + 3*log(G(3x)) + 9*log(G(9x))^2/2! + 27*log(G(27x))^3/3! + 81*log(G(81x))^4/4! +...
ILLUSTRATE (3^n)-th POWERS OF G.F. G(x) OF A167003.
The coefficients in the expansion of G(x)^(3^n), n>=0, begin:
G^1: [1, 1, -4, -4, -8220, -16910960, -220513689396,...];
G^3: [(1), 3, -9, -35, -24648, -50782068, -661642361248,...];
G^9: [1, (9), 0, -240, -74574, -152788194, -1985840486856,...];
G^27: [1, 27, (243), 9, -236682, -462449898, -5965789971726,...];
G^81: [1, 81, 2916, (59076), 0, -1420876404, -17973134801100,...];
G^243: [1, 243, 28431, 2125845, (111615732), 0, -54490964413644,...];
G^729: [1, 729, 262440, 62178840, 10895760846, (1491228760410), 0,...]; ...
where the coefficients along the diagonal (shown in parenthesis) form the initial terms of this sequence.
PROG
(PARI) {a(n)=local(A=[1, 9], B=[1, 3], G=[1, 1]); for(i=1, n, G=concat(G, 0); B=Vec(sum(m=0, #G, log(subst(Ser(G), x, 3^m*x))^m/m!)); G[ #G]=-floor(B[ #G]/3^(#G-1))); A=Vec(sum(m=0, #G, 3^m*log(subst(Ser(G), x, 3^m*x))^m/m!)); A[n+1]}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 14 2009
STATUS
approved