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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":"c33bb1fa30bb8ff8d17e2600188f3ff881c64d1fbe3a59922b683cedd11e70c9","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-edcd820194369f8f","role":"sequence","name":"basis_map_A013611 of Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).","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:     276,      480,       192;\n  4:    1640,     4512,      3840,      1024;\n  5:    9008,    34240,     45696,     25600,      5120;\n  6:   47160,   229728,    422400,    368640,    153600,     24576;\n  7:  239068,  1423200,   3353280,   4019200,   2595840,    860160,    114688;\n  8: 1184200,  8339360,  24038400,  36874240,  32614400,  16687104,   4587520,   524288;\n  9: 5764620, 46900800, 160168320, 300518400, 339655680, 237404160, 100466688, 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) + (13+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] + (13 + 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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