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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":"57c46b93707d5152e5127b82c6ed95e431595594c588afe748ac535055483e9e","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-de4284a2d5d7d0a6","role":"sequence","name":"A Motzkin related 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: 1;\n  1: 1, 1;\n  2: 1, 3,  1;\n  3: 0, 6,  6,   1;\n  4: 0, 4, 20,  10,   1;\n  5: 0, 0, 30,  50,  15,    1;\n  6: 0, 0, 15, 125, 105,   21,    1;\n  7: 0, 0,  0, 140, 385,  196,   28,   1;\n  8: 0, 0,  0,  56, 714,  980,  336,  36,  1;\n  9: 0, 0,  0,   0, 630, 2646, 2184, 540, 45, 1;\n","recurrence_sentence":"The row polynomials satisfy -(n^3+n^2-2*n-(n^3-3*n^2-4*n+12)*x-(2*n^2-2*n-12)*x^2-(n+2)*x^3)*P(n,x) = (4*n^3-16*n^2+12*n-(7*n^3-27*n^2+38*n-24)*x+(3*n^3-21*n^2+36*n-24)*x^2+(6*n^2-19*n+4)*x^3+3*n*x^4)*P(n-1,x) - ((8*n^2-8*n)*x-(10*n^2-30*n+16)*x^2+(2*n^2-23*n+36)*x^3+(4*n-16)*x^4+2*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^3 - 16*n^2 + 12*n - (7*n^3 - 27*n^2 + 38*n - 24)*x + (3*n^3 - 21*n^2 + 36*n - 24)*x^2 + (6*n^2 - 19*n + 4)*x^3 + 3*n*x^4)*P[n - 1, x] - ((8*n^2 - 8*n)*x - (10*n^2 - 30*n + 16)*x^2 + (2*n^2 - 23*n + 36)*x^3 + (4*n - 16)*x^4 + 2*x^5)*D[P[n - 1, x], x])/(-n^3 - n^2 + 2*n + (n^3 - 3*n^2 - 4*n + 12)*x + (2*n^2 - 2*n - 12)*x^2 + (n + 2)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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