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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":"eb76e957745c7d5a8df81b63782f4e1ac54061f4aaf33968db02da48ce20c11f","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-89d60963344cadbc","role":"sequence","name":"basis_map_A013610 of Triangular array formed by the Mersenne numbers.","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:     24,      18;\n  3:    132,     216,       81;\n  4:    624,    1620,     1296,      324;\n  5:   2724,    9792,    12312,     6480,     1215;\n  6:  11328,   52272,    90720,    74520,    29160,     4374;\n  7:  45600,  257472,   573480,   648000,   393660,   122472,   15309;\n  8: 179328, 1198944,  3271104,  4736880,  3946320,  1898316,  489888,   52488;\n  9: 692976, 5356800, 17332704, 30735936, 32761260, 21555072, 8573040, 1889568, 177147;\n","recurrence_sentence":"The row polynomials satisfy (6*n-6)*P(n,x) = (12*n+24+(9*n+18)*x)*P(n-1,x) + (8+24*x+9*x^2)*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[((12*n + 24 + (9*n + 18)*x)*P[n - 1, x] + (8 + 24*x + 9*x^2)*D[P[n - 1, x], x])/(6*n - 6)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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