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 A177893 Irregular triangle read by rows giving coefficients of Yablonskii-Vorob'ev polynomials. 1
 4, 0, 0, 1, -80, 0, 0, 20, 1, 0, 11200, 0, 0, 0, 0, 60, 0, 0, 1, -6272000, 0, 0, -3136000, 0, 0, 78400, 0, 0, 2800, 0, 0, 140, 0, 0, 1, -38635520000, 0, 0, 19317760000, 0, 0, 1448832000, 0, 0, -17248000, 0, 0, 627200, 0, 0, 18480, 0, 0, 280, 0, 0, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS From Table 1, p.2 of Roffelsen. Yablonskii-Vorob'ev polynomials are special polynomials used to represent rational solutions of the second Painlev'e equation. The Yablonskii-Vorob'ev polynomials are defined by the differential-difference equation Q_(n+1)Q_(n-1) = z*(Q_n)^2 - 4*(Q_n * (Q_n)'' - ((Q_n)')^2, with Q_0 = 1 and Q_1 = z. Divisibility properties of the coefficients of these polynomials, concerning powers of 4, are obtained and we prove that the nonzero roots of the Yablonskii-Vorob'ev polynomials are irrational. Furthermore, relations between the roots of these polynomials for consecutive degree are found by considering power series expansions of rational solutions of the second Painlev'e equation. Since The Yablonskii-Vorob_ev polynomials Q_n are monic polynomials of degree n*(n + 1)/2 = A000217(n), with integer coefficients, each row of this table ends with 1. LINKS Pieter Roffelsen, Irrationality of the Roots of the Yablonskii-Vorob'ev Polynomials and Relations between Them, Dec 14, 2010. EXAMPLE Q_2 = 4 + z^3 so row 2 of the table is 4, 0, 0, 1 denoting 4*z^0 + 0*z^1 + 0*z^2 + 1*z^3. Q_3 = -80 + 20*z^3 + 1*z^6 so row 3 of the table is -80, 0, 0, 20, 1. Q_4 = z(11200 + 60z6 + z9)= 11200*z + 60*z^7 + 1*z10, so row 4 of the table is 0, 11200, 0, 0, 0, 0, 60, 0, 0, 1. Q_5 = -6272000 - 3136000*z^3 + 78400*z^6 + 2800*z^9 + 140*z^12 + 1*z^15, so row 5 of the table is -6272000, 0, 0, -3136000, 0, 0, 78400, 0, 0, 2800, 0, 0, 140, 0, 0, 1. Q_6 = -38635520000 + 19317760000*z^3 + 1448832000*z^6 - 17248000*z^9 + 627200*z^12 + 18480*z^15 + 280*z^18 + 1*z^21, so row 6 of the table is -38635520000, 0, 0, 19317760000, 0, 0,  1448832000, 0, 0, -17248000, 0, 0, 627200, 0, 0, 18480, 0, 0, 280, 0, 0, 1. Q_7 =z*(-3093932441600000 - 49723914240000*z^6 - 828731904000*z^9 + 13039488000*z^12 + 62092800*z^15 + 5174400*z^18 + 75600*z^21 + 504*z^24 + 1*z^27). Q_8 = -991048439693312000000 - 743286329769984000000*^z^3 + 37164316488499200000*z^6 + 1769729356595200000*z^9 + 126696533483520000*z^12 + 407736096768000*z^15 - 6629855232000*z^18 + 124309785600*z^21 + 2018016000*z^24 + 32771200*z^27 + 240240*z^30 + 840*z^33 + 1*z^36. CROSSREFS Cf. A000217. Sequence in context: A194794 A317448 A292900 * A058305 A193717 A020808 Adjacent sequences:  A177890 A177891 A177892 * A177894 A177895 A177896 KEYWORD sign,tabf AUTHOR Jonathan Vos Post, Dec 14 2010 STATUS approved

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Last modified October 1 04:06 EDT 2020. Contains 337441 sequences. (Running on oeis4.)