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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":"78ba5a852b31c1634152af8d4a4f0aff6b0a2182b0ff06170ba0c04601720d88","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-497753d6192915db","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of permutations of an n-set having k cycles of size > 1 (0<=k<=floor(n/2)).","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;\n  2:      2,      1;\n  3:      6,      5;\n  4:     21,     26,      3;\n  5:     85,    154,     35;\n  6:    410,    999,    340,    15;\n  7:   2366,   7083,   3304,   315;\n  8:  16065,  55404,  33110,  4900,  105;\n  9: 125665, 479820, 347094, 70532, 3465;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2-4*n+3-(2*n^2-8*n+6)*x+(n^2-4*n+3)*x^2)*P(n-4,x) + (2*n^2-6*n+4-(2*n^2-6*n+4)*x)*P(n-3,x) - (n^2-1-(2*n-2)*x)*P(n-2,x) + (2*n-1)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 2 + x, P(3,x) = 6 + 5*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[2, x_] := 2 + x;\nP[3, x_] := 6 + 5*x;\nP[n_, x_] := P[n, x] = Expand[-(n^2 - 4*n + 3 - (2*n^2 - 8*n + 6)*x + (n^2 - 4*n + 3)*x^2)*P[n - 4, x] + (2*n^2 - 6*n + 4 - (2*n^2 - 6*n + 4)*x)*P[n - 3, x] - (n^2 - 1 - (2*n - 2)*x)*P[n - 2, x] + (2*n - 1)*P[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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