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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":"2d172527f33500622846ef5b16265bf05a0c8e8e8f89e7f6c55e3d60fb387dde","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-9ed87ca196739b1c","role":"sequence","name":"Triangle read by rows, where the g.f. satisfies A(x, y) = 1 + x*A(x, y)^2 + x*y*A(x, y)^3.","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":[{"id":"l-ae5da44dafde2f7e","kind":"fingerprint_related","node_a_id":"n-d95c619ac7adfec9","node_b_id":"n-9b6812a2cbd6bf83","fingerprint":"f6-c25c6caf35317535","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c25c6caf35317535"},"other_id":"n-9b6812a2cbd6bf83","other":{"id":"n-9b6812a2cbd6bf83","name":"derivative of Triangle read by rows, interpolating between the central binomial coefficients and the central coefficients of the Catalan triangle. 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