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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":"5309ee5ab284eaa8c1fbefe2d254bdba83c90618951cd32f7a4f52627220a763","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-fe801e460afa1e3b","role":"sequence","name":"basis_map_A013609 of Square the entries of Pascal's triangle.","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:      2;\n  2:     12,       8;\n  3:     60,      96,       24;\n  4:    280,     720,      480,       64;\n  5:   1260,    4480,     5040,     1920,      160;\n  6:   5544,   25200,    40320,    26880,     6720,     384;\n  7:  24024,  133056,   277200,   268800,   120960,   21504,     896;\n  8: 102960,  672672,  1729728,  2217600,  1478400,  483840,   64512,   2048;\n  9: 437580, 3294720, 10090080, 16144128, 14414400, 7096320, 1774080, 184320, 4608;\n","recurrence_sentence":"The row polynomials satisfy (n^2-2*n+1)*P(n,x) = (4*n^2-6*n+2+(6*n^2-12*n+4)*x)*P(n-1,x) - ((2*n-1)*x+(4*n-2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 6*n + 2 + (6*n^2 - 12*n + 4)*x)*P[n - 1, x] - ((2*n - 1)*x + (4*n - 2)*x^2)*D[P[n - 1, x], x])/(n^2 - 2*n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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