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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":"d084bad28533c41657cef90bea75b2715d73faadeedd58638421ce7caedeee7e","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-817c504263a54deb","role":"sequence","name":"reversal of Coefficient triangle of certain polynomials N(5; m,x).","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:    14,      2;\n  3:    84,     48,      3;\n  4:   336,    432,    108,       4;\n  5:  1050,   2400,   1350,     200,      5;\n  6:  2772,   9900,   9900,    3300,    330,      6;\n  7:  6468,  33264,  51975,   30800,   6930,    504,     7;\n  8: 13728,  96096, 216216,  200200,  80080,  13104,   728,    8;\n  9: 27027, 247104, 756756, 1009008, 630630, 183456, 22932, 1008, 9;\n","recurrence_sentence":"The row polynomials satisfy (n^3+3*n^2-9*n+5)*P(n,x) = (n^3+10*n^2+25*n+(3*n^3+4*n^2-13*n)*x)*P(n-1,x) + ((2*n^2+4*n)*x-(2*n^2+4*n)*x^2)*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[((n^3 + 10*n^2 + 25*n + (3*n^3 + 4*n^2 - 13*n)*x)*P[n - 1, x] + ((2*n^2 + 4*n)*x - (2*n^2 + 4*n)*x^2)*D[P[n - 1, x], x])/(n^3 + 3*n^2 - 9*n + 5)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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