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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":"ee29dec2acad8f26365ad6b2e9732e085cb8bd4f285a624f8682e0b88bec3af7","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-1f0d56c568ab8221","role":"sequence","name":"basis_map_A007318 of Riordan array (1, 3+3x).","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:      21,      18;\n  3:     117,     198,       81;\n  4:     585,    1476,     1215,      324;\n  5:    2754,    9234,    11421,     6156,     1215;\n  6:   12474,   52164,    86103,    69984,    27945,     4374;\n  7:   54999,  275562,   568863,   618516,   373005,   118098,   15309;\n  8:  237735, 1388016,  3439665,  4685364,  3784725,  1810836,  474579,   52488;\n  9: 1012095, 6747138, 19511685, 31945752, 32362740, 20754630, 8220933, 1837080, 177147;\n","recurrence_sentence":"The row polynomials satisfy (n^2-4*n+3-(3*n-3)*x)*P(n,x) = -(2*n^2-23*n+45-(15*n-57)*x+18*x^2)*P(n-1,x) + (7*n-28+(10*n-61)*x+(3*n-42)*x^2-9*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[(-(2*n^2 - 23*n + 45 - (15*n - 57)*x + 18*x^2)*P[n - 1, x] + (7*n - 28 + (10*n - 61)*x + (3*n - 42)*x^2 - 9*x^3)*D[P[n - 1, x], x])/(n^2 - 4*n + 3 - (3*n - 3)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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