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%I #26 Mar 19 2024 20:01:15
%S 1,0,1,4,0,0,1,-80,0,0,20,1,0,11200,0,0,0,0,60,0,0,1,-6272000,0,0,
%T -3136000,0,0,78400,0,0,2800,0,0,140,0,0,1,-38635520000,0,0,
%U 19317760000,0,0,1448832000,0,0,-17248000,0,0,627200,0,0,18480,0,0,280,0,0,1
%N Irregular triangle read by rows giving coefficients of Yablonskii-Vorob'ev polynomials.
%C From Table 1, p.2 of Roffelsen. The Yablonskii-Vorob'ev polynomials are defined by the 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.
%C Q_n has degree n*(n + 1)/2 = A000217(n).
%H Pieter Roffelsen, <a href="http://arxiv.org/abs/1012.2933">Irrationality of the Roots of the Yablonskii-Vorob'ev Polynomials and Relations between Them</a>, arXiv:1012.2933 [math.CA], Dec 14 2010.
%e 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.
%e Q_3 = -80 + 20*z^3 + 1*z^6 so row 3 of the table is -80, 0, 0, 20, 1.
%e Q_4 = z(11200 + 60z^6 + z^9)= 11200*z + 60*z^7 + 1*z^10, so row 4 of the table is 0, 11200, 0, 0, 0, 0, 60, 0, 0, 1.
%e 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.
%e Q_6 = -38635520000 + 19317760000*z^3 + 1448832000*z^6 - 17248000*z^9 + 627200*z^12
%e + 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,
%e 18480, 0, 0, 280, 0, 0, 1.
%e 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).
%e 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.
%t r[q_, f_, z_] := z q^2 + f (q D[q, z, z] - D[q, z]^2);
%t a[f_, n_] := Module[{t = {{1}}, p = 1, q = 1, z},
%t Do[{p, q} = {q, Simplify[r[q, f, z]/p]}; AppendTo[t, CoefficientList[q, z]], n];
%t t];
%t Flatten[a[-4, 6]] (* this sequence *)
%t Flatten[a[1, 6][[;;,-1;;1;;-3]]] (* A092766 *)
%t (* _Andrey Zabolotskiy_, Sep 29 2021 *)
%Y Cf. A000217, A092766.
%K sign,tabf
%O 0,4
%A _Jonathan Vos Post_, Dec 14 2010
%E Rows 0, 1 added by _Andrey Zabolotskiy_, Sep 29 2021
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