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 A167004 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. 3
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Table of n, a(n) for n=0..80. FORMULA 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. EXAMPLE 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'). PROG (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]} CROSSREFS Cf. A167003, A167005, variant: A167001. Sequence in context: A192919 A068607 A303633 * A287632 A259346 A239798 Adjacent sequences: A167001 A167002 A167003 * A167005 A167006 A167007 KEYWORD nonn AUTHOR Paul D. Hanna, Nov 14 2009 STATUS approved

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Last modified April 14 14:16 EDT 2024. Contains 371665 sequences. (Running on oeis4.)