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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":"38bc6f9a8303436578eb4804fffdbbb9aaccc2f5db92aa7ba48e2c35fa75ae98","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-e646c30d067149e7","role":"sequence","name":"Triangular array:  row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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:  1;\n  1:  5,    4;\n  2:  9,   24,     8;\n  3: 13,   76,    80,    16;\n  4: 17,  160,   360,   224,     32;\n  5: 21,  276,  1040,  1328,    576,     64;\n  6: 25,  424,  2312,  4928,   4256,   1408,    128;\n  7: 29,  604,  4368, 13648,  19392,  12480,   3328,   256;\n  8: 33,  816,  7400, 31136,  64800,  67584,  34432,  7680,   512;\n  9: 37, 1060, 11600, 62064, 174912, 265536, 216320, 90880, 17408, 1024;\n","recurrence_sentence":"The row polynomials satisfy -(4*n^3+n^2-3*n-(8*n^3-10*n^2-3*n+15)*x-(20*n^2-38*n+10)*x^2-(8*n^2-4*n-12)*x^3-(8*n-8)*x^4)*P(n,x) = -(4*n^3+5*n^2+n+(24*n^3-6*n^2-45*n-15)*x+(44*n^2-34*n-10)*x^2+(24*n^2+36*n+12)*x^3+(24*n+8)*x^4)*P(n-1,x) + ((4*n^2+5*n+1)*x+(48*n^2+72*n+19)*x^2+(16*n^2+168*n+100)*x^3+(96*n+176)*x^4+(16*n+96)*x^5+16*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 + 5*n^2 + n + (24*n^3 - 6*n^2 - 45*n - 15)*x + (44*n^2 - 34*n - 10)*x^2 + (24*n^2 + 36*n + 12)*x^3 + (24*n + 8)*x^4)*P[n - 1, x] + ((4*n^2 + 5*n + 1)*x + (48*n^2 + 72*n + 19)*x^2 + (16*n^2 + 168*n + 100)*x^3 + (96*n + 176)*x^4 + (16*n + 96)*x^5 + 16*x^6)*D[P[n - 1, x], x])/(-4*n^3 - n^2 + 3*n + (8*n^3 - 10*n^2 - 3*n + 15)*x + (20*n^2 - 38*n + 10)*x^2 + (8*n^2 - 4*n - 12)*x^3 + (8*n - 8)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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