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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":"0b587cd6c26b55ed76380bfff2697069139e21a29c87a766145ae8d06864ebd9","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-e3a9cbbad5cb7ab4","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of words over alphabet {0,1,2,3} having exactly k occurrences of the string 01, where n>=0 and k>=0.","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:     0;\n  2:     1;\n  3:     8;\n  4:    46,     2;\n  5:   232,    24;\n  6:  1091,   186,    3;\n  7:  4912,  1184,   48;\n  8: 21468,  6732,  468,  4;\n  9: 91824, 35568, 3600, 80;\n","recurrence_sentence":"The row polynomials satisfy (2*n-4)*P(n,x) = (7*n-5+(n-3)*x)*P(n-1,x) + (6-4*x-2*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((7*n - 5 + (n - 3)*x)*P[n - 1, x] + (6 - 4*x - 2*x^2)*D[P[n - 1, x], x])/(2*n - 4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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