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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":"4611ce04dad8f47eaef6cfa90777565169fec0b6f52f2d931d60e6c654fa6454","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-ca41bee2bc8fcdbe","role":"sequence","name":"The convolution matrix of the double factorial of odd numbers (A001147).","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:          1;\n  1:          4,          1;\n  2:         24,         11,          1;\n  3:        192,        123,         21,         1;\n  4:       1920,       1545,        375,        34,        1;\n  5:      23040,      22005,       6795,       885,       50,       1;\n  6:     322560,     353115,     130725,     21840,     1785,      69,      1;\n  7:    5160960,    6325515,    2706165,    545685,    57540,    3234,     91,    1;\n  8:   92897280,  125345745,   60448815,  14165550,  1811565,  131922,   5418,  116,   1;\n  9: 1857945600, 2725157925, 1455219675, 386256465, 57643110, 5109615, 272790, 8550, 144, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2*n+2+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(2*n + 2 + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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