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b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"d320d54ceaa02d9e6977807bea43b43860819094954b550f877fd1b04efa115f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-8293244ef059e34c","role":"sequence","name":"Triangle with T(n,k) = k*E(n,k) where E(n,k) are Eulerian numbers A008292.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1,   10,      12;\n   2: 1,   25,      80,       60;\n   3: 1,   56,     360,      660,       360;\n   4: 1,  119,    1372,     4620,      5880,       2520;\n   5: 1,  246,    4788,    26376,     58800,      57120,       20160;\n   6: 1,  501,   15864,   134316,    466704,     771120,      604800,      181440;\n   7: 1, 1012,   50880,   637020,   3238200,    8094240,    10584000,     6955200,     1814400;\n   8: 1, 2035,  159764,  2882220,  20603880,   73097640,   141704640,   153014400,    86486400,    19958400;\n   9: 1, 4082,  494604, 12623952, 123552000,  597738240,  1611308160,  2542034880,  2335132800,  1157587200,   239500800;\n  10: 1, 8177, 1516528, 54031692, 710231808, 4559480640, 16416354240, 35485369920, 47100493440, 37621584000, 16605388800, 3113510400;\n","recurrence_sentence":"The row polynomials satisfy (1+3*x+3*x^2+x^3)*P(n,x) = -((n+2)*x^2+(3*n+6)*x^3+(3*n+6)*x^4+(n+2)*x^5)*P(n-2,x) - ((n+2)*x^2+(4*n+8)*x^3+(6*n+12)*x^4+(4*n+8)*x^5+(n+2)*x^6)*d/dx P(n-2,x) + (1+(n+6)*x+(3*n+12)*x^2+(3*n+10)*x^3+(n+3)*x^4)*P(n-1,x) + (x+4*x^2+6*x^3+4*x^4+x^5)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1 + 10*x + 12*x^2, P(2,x) = 1 + 25*x + 80*x^2 + 60*x^3.","mathematica":"P[1, x_] := 1 + 10*x + 12*x^2;\nP[2, x_] := 1 + 25*x + 80*x^2 + 60*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-((n + 2)*x^2 + (3*n + 6)*x^3 + (3*n + 6)*x^4 + (n + 2)*x^5)*P[n - 2, x] - ((n + 2)*x^2 + (4*n + 8)*x^3 + (6*n + 12)*x^4 + (4*n + 8)*x^5 + (n + 2)*x^6)*D[P[n - 2, x], x] + (1 + (n + 6)*x + (3*n + 12)*x^2 + (3*n + 10)*x^3 + (n + 3)*x^4)*P[n - 1, x] + (x + 4*x^2 + 6*x^3 + 4*x^4 + x^5)*D[P[n - 1, x], x])/(1 + 3*x + 3*x^2 + x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1 + 10*x + 12*x^2; v[2] = 1 + 25*x + 80*x^2 + 60*x^3; for(n = 3, N, my(i = n); v[i] = (-((n + 2)*x^2 + (3*n + 6)*x^3 + (3*n + 6)*x^4 + (n + 2)*x^5)*v[i - 2] - ((n + 2)*x^2 + (4*n + 8)*x^3 + (6*n + 12)*x^4 + (4*n + 8)*x^5 + (n + 2)*x^6)*deriv(v[i - 2], x) + (1 + (n + 6)*x + (3*n + 12)*x^2 + (3*n + 10)*x^3 + (n + 3)*x^4)*v[i - 1] + (x + 4*x^2 + 6*x^3 + 4*x^4 + x^5)*deriv(v[i - 1], x))/(1 + 3*x + 3*x^2 + x^3)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-ff270bcb802a85af","steps":["\u03b8+1 (shift +2)","x \u21a6 x/(x+1)"],"edges":["e-fb10cc40ca2d4795","e-4f4a8f2fc12ad243"],"text":"A008292 \u2192 \u03b8+1 (shift +2) \u2192 x \u21a6 x/(x+1)"},"headline":"A008292 \u2192 \u03b8+1 (shift +2) \u2192 x \u21a6 x/(x+1)","chain":null}