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D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"d8d043fff6d78a2e2dda423cd2711d28a1f48183cf6792de52bc8da80b4e4d18","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-f7cf3d554e6e9f71","role":"sequence","name":"basis_map_A013611 of Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, 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":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:        0;\n  1:        4;\n  2:       40,       32;\n  3:      292,      480,       192;\n  4:     1880,     4704,      3840,      1024;\n  5:    11312,    38080,     47232,     25600,      5120;\n  6:    65240,   276192,    460800,    378880,    153600,     24576;\n  7:   365484,  1864800,   3913920,   4326400,   2657280,    860160,    114688;\n  8:  2004600, 11971680,  30297600,  42127360,  34764800,  17031168,   4587520,   524288;\n  9: 10819020, 74030400, 219404160, 367897600, 381803520, 251166720, 102301696, 23592960, 2359296;\n","recurrence_sentence":"The row polynomials satisfy (8*n-8)*P(n,x) = (20*n+40+(16*n+32)*x)*P(n-1,x) + (21+40*x+16*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[((20*n + 40 + (16*n + 32)*x)*P[n - 1, x] + (21 + 40*x + 16*x^2)*D[P[n - 1, x], x])/(8*n - 8)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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