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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":"c72c2a557f31ac1a370dbdef7cefd5a200894513b3f0c14056186340ed82078d","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-e8e64448d4c6498a","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of binary words of length n with k strictly increasing runs, for 0<=k<=n.","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: 2;\n  2: 1, 6;\n  3: 0, 8, 12;\n  4: 0, 2, 30, 20;\n  5: 0, 0, 18, 80,  30;\n  6: 0, 0,  3, 84, 175,  42;\n  7: 0, 0,  0, 32, 280, 336,   56;\n  8: 0, 0,  0,  4, 180, 756,  588,  72;\n  9: 0, 0,  0,  0,  50, 720, 1764, 960, 90;\n","recurrence_sentence":"The row polynomials satisfy -(n^2-n-(4*n^2-6*n+2)*x)*P(n,x) = (n^2-4*n+3-(7*n^2-14*n+3)*x+(12*n^2-14*n-2)*x^2)*P(n-1,x) - ((2*n-2)*x-(10*n-2)*x^2+8*n*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((n^2 - 4*n + 3 - (7*n^2 - 14*n + 3)*x + (12*n^2 - 14*n - 2)*x^2)*P[n - 1, x] - ((2*n - 2)*x - (10*n - 2)*x^2 + 8*n*x^3)*D[P[n - 1, x], x])/(-n^2 + n + (4*n^2 - 6*n + 2)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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