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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":"5d1990cd22ce0314a1eba187731d3a2563f82d4f89a6f828b2557435d4413cb1","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-8fd7b2e367875124","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","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:    12,      8;\n   4:    50,     72,     24;\n   5:   180,    408,    288,      64;\n   6:   600,   1872,   2064,     960,     160;\n   7:  1908,   7608,  11520,    8320,    2880,    384;\n   8:  5878,  28584,  55320,   54720,   29280,   8064,    896;\n   9: 17700, 101656, 240192,  303360,  221760,  94080,  21504,  2048;\n  10: 52390, 347184, 970224, 1495872, 1394400, 806400, 283136, 55296, 4608;\n","recurrence_sentence":"The row polynomials satisfy (4*n-8)*P(n,x) = (6*n+6+(4*n+4)*x)*P(n-1,x) + (5+12*x+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 + 6 + (4*n + 4)*x)*P[n - 1, x] + (5 + 12*x + 4*x^2)*D[P[n - 1, x], x])/(4*n - 8)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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