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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":"0b1460f931d96492730817c021d437626b8267607a48f8b7a8fdad1166e9fe73","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-5aab339818c7e39d","role":"sequence","name":"basis_map_A013610 of A convolution triangle of numbers based on A001906 (even-indexed Fibonacci 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:     138,     216,       81;\n  4:     696,    1674,     1296,      324;\n  5:    3273,   10656,    12636,     6480,     1215;\n  6:   14736,   60588,    97200,    76140,    29160,     4374;\n  7:   64404,  320112,   648810,   686880,   400950,   122472,   15309;\n  8:  275472, 1606284,  3938544,  5266620,  4150440,  1928934,  489888,   52488;\n  9: 1159119, 7757856, 22333644, 36133776, 35950635, 22534848, 8695512, 1889568, 177147;\n","recurrence_sentence":"The row polynomials satisfy (2*n-2)*P(n,x) = (4*n+8+(3*n+6)*x)*P(n-1,x) + (4+8*x+3*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[((4*n + 8 + (3*n + 6)*x)*P[n - 1, x] + (4 + 8*x + 3*x^2)*D[P[n - 1, x], x])/(2*n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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