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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":"f995056f3442bbe48cf04a75f932e31080c51be2ce3b107d5c53110b4d90a09b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-7d7dad3bd149bf81","role":"sequence","name":"basis_map_A013609 of A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).","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:      2;\n  2:     16,       8;\n  3:     92,      96,      24;\n  4:    464,     744,     384,      64;\n  5:   2182,    4736,    3744,    1280,     160;\n  6:   9824,   26928,   28800,   15040,    3840,     384;\n  7:  42936,  142272,  192240,  135680,   52800,   10752,    896;\n  8: 183648,  713904, 1166976, 1040320,  546560,  169344,  28672,  2048;\n  9: 772746, 3447936, 6617376, 7137536, 4734240, 1978368, 508928, 73728, 4608;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (2*n+4+(n+2)*x)*P(n-1,x) + (3+4*x+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 + 4 + (n + 2)*x)*P[n - 1, x] + (3 + 4*x + x^2)*D[P[n - 1, x], x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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