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A143105 Let g_0(x,y)=x, g_1(x,y)=x^3+3xy and g_{n+2}(x,y) = (x^2+2y)g_{n+1}(x+y)-y^2g_n(x,y). The entries of the sequence are those odd d for which g_d(x,y) and cx^jy^kg_m(x,y) have at least two terms in common (same coefficients) for some c > 0 and integers j,k and such that g_d(x,y) + cx^jy^k(1+y^m - g_m(x,y)) has all positive coefficients. 1

%I

%S 1,3,5,9,17,21,33,41,45,53,69,77,81,93,105,113,117,125,129,141,149,

%T 153,161,165,177,185,201,213,221,225,249,261,269,273,285,297,305,309,

%U 333,341,345,357,365,369,381,405,413,417,429,437,441,453,465,473,489,501

%N Let g_0(x,y)=x, g_1(x,y)=x^3+3xy and g_{n+2}(x,y) = (x^2+2y)g_{n+1}(x+y)-y^2g_n(x,y). The entries of the sequence are those odd d for which g_d(x,y) and cx^jy^kg_m(x,y) have at least two terms in common (same coefficients) for some c > 0 and integers j,k and such that g_d(x,y) + cx^jy^k(1+y^m - g_m(x,y)) has all positive coefficients.

%C Note that g_k(x,y) always has positive coefficients. The sequence are degrees for which a certain construction (see paper by D'Angelo-Lebl) of proper monomial holomorphic mappings of balls does not give new noninvariant monomial mappings.

%C It is unknown if this sequence is infinite (conjectured to be so). Furthermore A143106 is definitely a subsequence of this sequence, but it is unknown if the two are in fact equal.

%D J. P. D'Angelo, Simon Kos and Emily Riehl. A sharp bound for the degree of proper monomial mappings between balls. J. Geom. Anal., 13(4) (2003) 581-593.

%H J. P. D'Angelo and J. Lebl. <a href="http://arXiv.org/abs/0708.3232">Complexity results for CR mappings between spheres</a>, arXiv:0708.3232, <a href="http://dx.doi.org/10.1142/S0129167X09005248">Internat. J. Math. 20 (2) (2009) 149</a>

%H J. Lebl and D. Lichtblau, <a href="http://arxiv.org/abs/0808.0284">Uniqueness of certain polynomials constant on a hyperplane</a>, arXiv:0808.0284 (2008).

%e For example when d=7, we get the following new polynomials x^7 + 7/2 xy + 7/2 x^5y 7/2 xy^5 and x^7 + 7 x^3y + 7 xy^3 + 7 x^3y^3. Hence 7 is not in the sequence.

%t See the paper by Lebl-Lichtblau

%Y Cf. A143106.

%K nonn

%O 0,2

%A Jiri Lebl (jlebl(AT)math.uiuc.edu), Jul 25 2008

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