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A167004
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Least possible nonnegative coefficients of x^n in G(x)^(3^n), n>=0, such that G(x) is an integer series with G'(0)=G(0)=1; the G(x) that satisfies this condition equals the g.f. of A167003.
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3
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1, 3, 0, 9, 0, 0, 0, 0, 0, 2187, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 282429536481, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
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OFFSET
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0,2
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LINKS
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FORMULA
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G.f.: A(x) = 1 + Sum_{n>=0} 3^(3^n-n)*x^(3^n).
G.f.: A(x) = Sum_{n>=0} log(G(3^n*x))^n/n! where G(x) = g.f. of A167003.
a(n) = [x^n] G(x)^(3^n) for n>=0 where G(x) = g.f. of A167003.
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EXAMPLE
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G.f.: A(x) = 1 + 3*x + 9*x^3 + 2187*x^9 + 282429536481*x^27 +...
A(x) = 1 + 3^(1-0)*x + 3^(3-1)*x^3 + 3^(9-2)*x^9 + 3^(27-3)*x^27 + 3^(81-4)*x^81 +...
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 + log(G(3x)) + log(G(9x))^2/2! + log(G(27x))^3/3! + 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 and equal 3^(3^m-m) at positions n=3^m for m>=0, with zeros elsewhere (except for the initial '1').
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PROG
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(PARI) {a(n)=if(n==0, 1, if(n==3^valuation(n, 3), 3^(n-valuation(n, 3)), 0))}
(PARI) /* A(x) = Sum_{n>=0} log(G(3^n*x))^n/n!, G(x) = g.f. of A167003: */ {a(n)=local(A=[1, 3], G=[1, 1]); for(i=1, n, G=concat(G, 0); A=Vec(sum(m=0, #G, log(subst(Ser(G), x, 3^m*x))^m/m!)); G[ #G]=-floor(A[ #G]/3^(#G-1))); A[n+1]}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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