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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":"51c22370f2d4436c6f29dc57176dc8c53f4012e0dc3896e9dd3ac1d08cd2b93e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-016d7724d51f6416","role":"sequence","name":"Riordan array (1, 3+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: 3;\n  2: 3, 18;\n  3: 0, 36,  81;\n  4: 0, 18, 243,  324;\n  5: 0,  0, 243, 1296,  1215;\n  6: 0,  0,  81, 1944,  6075,  4374;\n  7: 0,  0,   0, 1296, 12150, 26244,  15309;\n  8: 0,  0,   0,  324, 12150, 65610, 107163,  52488;\n  9: 0,  0,   0,    0,  6075, 87480, 321489, 419904, 177147;\n","recurrence_sentence":"The row polynomials satisfy (n^2-n-(3*n-3)*x)*P(n,x) = -(2*n^2-8*n+6-(15*n-21)*x+18*x^2)*P(n-1,x) + ((4*n-4)*x+(3*n-15)*x^2-9*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 8*n + 6 - (15*n - 21)*x + 18*x^2)*P[n - 1, x] + ((4*n - 4)*x + (3*n - 15)*x^2 - 9*x^3)*D[P[n - 1, x], x])/(n^2 - n - (3*n - 3)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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