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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":"84806aace72be29023af314f295a1e1dabb1494054a8a25263afbb0c73ce3113","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-01439b9265fd744f","role":"sequence","name":"basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1).","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-5a0e0e8c4a44e46c","kind":"oeis_match","node_a_id":"n-64e6cf669172ed21","node_b_id":null,"fingerprint":null,"oeis_id":"A007290","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A007290","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A007290","relation":"match_row_sums"}],"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:    2,     6;\n  3:   -8,    16,   12;\n  4:  -10,   -30,   60,    20;\n  5:   30,  -120,  -30,   160,     30;\n  6:   42,   126, -588,   140,    350,    42;\n  7: -112,   672, -168, -1792,    840,   672,   56;\n  8: -168,  -504, 4032, -3360,  -3780,  2772, 1176,   72;\n  9:  420, -3360, 2520, 13440, -16800, -5040, 7140, 1920, 90;\n","recurrence_sentence":"The row polynomials satisfy (n^2+3*n-4)*P(n,x) = -(2*n^2-2*n-10-(3*n^2+4*n-2)*x)*P(n-1,x) - (4*n+6-(6*n+9)*x+(2*n+3)*x^2)*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[(-(2*n^2 - 2*n - 10 - (3*n^2 + 4*n - 2)*x)*P[n - 1, x] - (4*n + 6 - (6*n + 9)*x + (2*n + 3)*x^2)*D[P[n - 1, x], x])/(n^2 + 3*n - 4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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