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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":"b4e28e15f7384d68f212a2f73cf798bad946904a2a2fc0361b78ada67658a049","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-ba196dcd01800741","role":"sequence","name":"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: 1;\n  2: 2, 2;\n  3: 0, 8,  3;\n  4: 0, 8, 18,   4;\n  5: 0, 0, 36,  32,   5;\n  6: 0, 0, 24,  96,  50,   6;\n  7: 0, 0,  0, 128, 200,  72,   7;\n  8: 0, 0,  0,  64, 400, 360,  98,   8;\n  9: 0, 0,  0,   0, 400, 960, 588, 128, 9;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-2*n-(n-1)*x)*P(n,x) = -(8*n^2-32*n+24-(10*n-14)*x+2*x^2)*P(n-1,x) + ((16*n-16)*x+(2*n-10)*x^2-x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(8*n^2 - 32*n + 24 - (10*n - 14)*x + 2*x^2)*P[n - 1, x] + ((16*n - 16)*x + (2*n - 10)*x^2 - x^3)*D[P[n - 1, x], x])/(2*n^2 - 2*n - (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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