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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":"b3e6b79e4183a0b05688f83ff163006d4d15acd30de16630c43aa23922a677dc","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-f364085508d20047","role":"sequence","name":"Triangle T(n,k) read by rows; related to number of preorders.","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: 2,  2;\n  2: 3, 15,   6;\n  3: 0, 40,  92,    24;\n  4: 0, 30, 405,   610,    120;\n  5: 0,  0, 630,  3906,   4524,     720;\n  6: 0,  0, 315,  9625,  38920,   37548,     5040;\n  7: 0,  0,   0, 10080, 134680,  411776,   346464,    40320;\n  8: 0,  0,   0,  3780, 221445, 1866060,  4667292,  3526416,   362880;\n  9: 0,  0,   0,     0, 173250, 4266570, 26457200, 56774520, 39296160, 3628800;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2-1)*x^2*P(n-4,x) - (2*n^2+n-1)*x^2*P(n-3,x) + ((2*n+2)*x-(n^2+n)*x^2)*P(n-2,x) + (2*n+2)*x*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + 2*x, P(2,x) = 3 + 15*x + 6*x^2, P(3,x) = 40*x + 92*x^2 + 24*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 2*x;\nP[2, x_] := 3 + 15*x + 6*x^2;\nP[3, x_] := 40*x + 92*x^2 + 24*x^3;\nP[n_, x_] := P[n, x] = Expand[-(n^2 - 1)*x^2*P[n - 4, x] - (2*n^2 + n - 1)*x^2*P[n - 3, x] + ((2*n + 2)*x - (n^2 + n)*x^2)*P[n - 2, x] + (2*n + 2)*x*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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