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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":"26bf755d9d69686895ada66623f77cfca64d1a5a7febfae782e053c2693d4c56","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-ed9434bfd889c409","role":"sequence","name":"Pascal-(1,4,1) 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:  0;\n  1:  1;\n  2:  6,    2;\n  3: 11,   22,     3;\n  4: 16,   92,    48,     4;\n  5: 21,  212,   318,    84,     5;\n  6: 26,  382,  1188,   764,   130,     6;\n  7: 31,  602,  3033,  4044,  1505,   186,    7;\n  8: 36,  872,  6228, 14424, 10380,  2616,  252,   8;\n  9: 41, 1192, 11148, 38904, 48630, 22296, 4172, 328, 9;\n","recurrence_sentence":"The row polynomials satisfy -(5*n^2-14*n+9-(5*n^2-3*n-2)*x+(n-1)*x^2)*P(n,x) = -(5*n^2-9*n+4+(45*n^2-177*n+104)*x-(24*n-30)*x^2+2*x^3)*P(n-1,x) + ((5*n-4)*x+(90*n-73)*x^2+(5*n-22)*x^3-x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(5*n^2 - 9*n + 4 + (45*n^2 - 177*n + 104)*x - (24*n - 30)*x^2 + 2*x^3)*P[n - 1, x] + ((5*n - 4)*x + (90*n - 73)*x^2 + (5*n - 22)*x^3 - x^4)*D[P[n - 1, x], x])/(-5*n^2 + 14*n - 9 + (5*n^2 - 3*n - 2)*x - (n - 1)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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