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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":"42df5fa34166c57cf32cc74ced2f12cfb6c6ada661d72dc68687a35dd5341a48","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:  0;\n  1:  4;\n  2:  8,   16;\n  3: 12,   64,    48;\n  4: 16,  144,   288,   128;\n  5: 20,  256,   912,  1024,    320;\n  6: 24,  400,  2112,  4224,   3200,    768;\n  7: 28,  576,  4080, 12288,  16320,   9216,   1792;\n  8: 32,  784,  7008, 28800,  57600,  56064,  25088,  4096;\n  9: 36, 1024, 11088, 58368, 160320, 233472, 177408, 65536, 9216;\n","recurrence_sentence":"The row polynomials satisfy -(n-1-(2*n-2)*x)*P(n,x) = -(n+(6*n-12)*x-8*x^2)*P(n-1,x) + (x+12*x^2+4*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n + (6*n - 12)*x - 8*x^2)*P[n - 1, x] + (x + 12*x^2 + 4*x^3)*D[P[n - 1, x], x])/(-n + 1 + (2*n - 2)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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