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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":"fb335e721183daea10b9d0897c3ff6b47718e860c1c95aa36e936af03c1f8927","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-eac80c2bf8e566f5","role":"sequence","name":"Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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:  3;\n  2:  6,   6;\n  3:  9,  30,    9;\n  4: 12,  72,   72,    12;\n  5: 15, 132,  270,   132,    15;\n  6: 18, 210,  684,   684,   210,    18;\n  7: 21, 306, 1395,  2316,  1395,   306,   21;\n  8: 24, 420, 2484,  6000,  6000,  2484,  420,  24;\n  9: 27, 552, 4032, 13032, 19290, 13032, 4032, 552, 27;\n","recurrence_sentence":"The row polynomials satisfy -(n-1-(n-1)*x)*P(n,x) = -(n+(5*n-10)*x-2*x^2)*P(n-1,x) + (x+10*x^2+x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n + (5*n - 10)*x - 2*x^2)*P[n - 1, x] + (x + 10*x^2 + x^3)*D[P[n - 1, x], x])/(-n + 1 + (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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