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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":"c037d147dce9b374a2a19dbf193ed4d47f6876e38d7f5cdfbb90453785f66c93","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-6ad01dc6e705cdb6","role":"sequence","name":"Pascal-(1,5,1) array.","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:  2,    1;\n  2:  8,    9,     1;\n  3: 14,   39,    16,     1;\n  4: 20,  141,   118,    23,      1;\n  5: 26,  315,   580,   245,     30,     1;\n  6: 32,  561,  2050,  1655,    420,    37,     1;\n  7: 38,  879,  5176,  7925,   3690,   643,    44,    1;\n  8: 44, 1269, 10606, 27911,  22800,  7009,   914,   51,  1;\n  9: 50, 1731, 18988, 75653, 104406, 53695, 11936, 1233, 58, 1;\n","recurrence_sentence":"The row polynomials satisfy -(6*n^3-16*n^2+10*n-(6*n^3-25*n^2+52*n)*x-(20*n^2-49*n)*x^2-(n^2+18*n)*x^3-n*x^4)*P(n,x) = -(6*n^3-10*n^2+4*n+(66*n^3-227*n^2+95*n)*x+(214*n^2-540*n+351)*x^2+(11*n^2+179*n-199)*x^3+(10*n-31)*x^4-x^5)*P(n-1,x) + ((6*n^2-4*n)*x+(132*n^2-67*n-10)*x^2+(6*n^2+459*n-199)*x^3+(43*n+453)*x^4+(n+43)*x^5+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[(-(6*n^3 - 10*n^2 + 4*n + (66*n^3 - 227*n^2 + 95*n)*x + (214*n^2 - 540*n + 351)*x^2 + (11*n^2 + 179*n - 199)*x^3 + (10*n - 31)*x^4 - x^5)*P[n - 1, x] + ((6*n^2 - 4*n)*x + (132*n^2 - 67*n - 10)*x^2 + (6*n^2 + 459*n - 199)*x^3 + (43*n + 453)*x^4 + (n + 43)*x^5 + x^6)*D[P[n - 1, x], x])/(-6*n^3 + 16*n^2 - 10*n + (6*n^3 - 25*n^2 + 52*n)*x + (20*n^2 - 49*n)*x^2 + (n^2 + 18*n)*x^3 + n*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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