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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":"dab4470288fda490cb357f5e2e367f6b9ada174bcf0f84a294a136f52d7801c1","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-bec6dabccdd13354","role":"sequence","name":"theta_plus_one of Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 2,    2;\n   3: 2,   14,      6;\n   4: 2,   38,     90,      24;\n   5: 2,   86,    474,     624,      120;\n   6: 2,  182,   1890,    5400,     4800,      720;\n   7: 2,  374,   6666,   33360,    61800,    41040,      5040;\n   8: 2,  758,  22050,  175224,   546000,   735840,    388080,     40320;\n   9: 2, 1526,  70314,  839664,  3984120,  8799840,   9243360,   4032000,   362880;\n  10: 2, 3062, 219330, 3802200, 25956000, 85019760, 143730720, 123137280, 45722880, 3628800;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(4*x^2-2*x^3)*P(n-2,x) - (x^2-x^4)*d/dx P(n-2,x) + (1+5*x-x^2)*P(n-1,x) + (x+x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 2*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-(4*x^2 - 2*x^3)*P[n - 2, x] - (x^2 - x^4)*D[P[n - 2, x], x] + (1 + 5*x - x^2)*P[n - 1, x] + (x + x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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