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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":"2e9ebd011c5b176b923ddbb5ec67ddae771cee6e21eaedf601eb93964e759cfc","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-32899adace0e98a1","role":"sequence","name":"basis_map_A013611 of A Jacobsthal 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:    4;\n  2:   16,     32;\n  3:   44,    224,     192;\n  4:  120,    896,    1920,    1024;\n  5:  292,   3008,   10368,   13312,     5120;\n  6:  704,   9056,   43008,   91136,    81920,    24576;\n  7: 1628,  25376,  154176,  457728,   680960,   466944,   114688;\n  8: 3720,  67776,  501120, 1921024,  4044800,  4571136,  2523136,   524288;\n  9: 8340, 174336, 1520640, 7153664, 19589120, 31457280, 28442624, 13107200, 2359296;\n","recurrence_sentence":"The row polynomials satisfy (8*n^2-12*n+4-(16*n-16)*x)*P(n,x) = -(32*n^2-168*n+160-(160*n-288)*x+128*x^2)*P(n-1,x) + (18*n-27+(80*n-156)*x+(32*n-208)*x^2-64*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[(-(32*n^2 - 168*n + 160 - (160*n - 288)*x + 128*x^2)*P[n - 1, x] + (18*n - 27 + (80*n - 156)*x + (32*n - 208)*x^2 - 64*x^3)*D[P[n - 1, x], x])/(8*n^2 - 12*n + 4 - (16*n - 16)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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