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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":"7086fb99ba12b98cadbdc93d6cdc7e84fc37134f469c0083635dec636182cdf6","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-45b8d33479b2056d","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:     0;\n   2:     2;\n   3:    14,      8;\n   4:    62,     80,      24;\n   5:   230,    480,     312,      64;\n   6:   780,   2280,    2352,    1024,     160;\n   7:  2508,   9480,   13584,    9280,    3040,    384;\n   8:  7786,  36192,   66840,   63040,   32160,   8448,    896;\n   9: 23578, 130240,  295512,  358080,  251040, 102144,  22400,  2048;\n  10: 70090, 448840, 1210416, 1799232, 1616160, 900480, 304640, 57344, 4608;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2-14*n+12)*P(n,x) = (6*n^2-4*n+(4*n^2-4*n)*x)*P(n-1,x) + (5*n-5+(12*n-12)*x+(4*n-4)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 2.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((6*n^2 - 4*n + (4*n^2 - 4*n)*x)*P[n - 1, x] + (5*n - 5 + (12*n - 12)*x + (4*n - 4)*x^2)*D[P[n - 1, x], x])/(4*n^2 - 14*n + 12)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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