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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":"4e451e12df159db301fd1555827c4d60f6a1d960cd7e336b64abb8649f30da96","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-5c56442adc0376c2","role":"sequence","name":"Triangle read by rows, the Bell transform of the quartic factorial numbers A007696(n+1) without column 0.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:            1;\n   2:            6,            1;\n   3:           60,           17,           1;\n   4:          840,          315,          33,          1;\n   5:        15120,         6825,         975,         54,         1;\n   6:       332640,       172095,       30225,       2325,        80,        1;\n   7:      8648640,      4979205,     1018395,      97650,      4725,      111,       1;\n   8:    259459200,    163003995,    37567845,    4240845,    258300,     8610,     147,     1;\n   9:   8821612800,   5964599025,  1515446415,  194479110,  14056245,   594090,   14490,   188,   1;\n  10: 335221286400, 241440974775, 66582455625, 9489089925, 784841400, 39602115, 1231650, 22950, 234, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4*n-2+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(4*n - 2 + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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