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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":"1616565d6e29655f00917bea94ac12ca148fcc08a4aaa2d87e33a60f349a0ded","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-c856ca922af03076","role":"sequence","name":"Triangle read by rows: T(n,k) = T(n-1,k-1) + k^2*T(n-1,k), 1 < k <= n, T(n,1) = 1.","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:     5,        2;\n  3:    21,       28,         3;\n  4:    85,      294,        90,         4;\n  5:   341,     2816,      1881,       220,        5;\n  6:  1365,    26026,     34320,      8008,      455,       6;\n  7:  5461,   236964,    588159,    245960,    26390,     840,      7;\n  8: 21845,  2143598,   9765990,   6933212,  1257490,   72828,   1428,    8;\n  9: 87381, 19336072, 159471237, 186351620, 53936155, 5077560, 176358, 2280, 9;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(4+2*x+x^2)*P(n-2,x) - (5*x+4*x^2)*d/dx P(n-2,x) - (x^2+x^3)*d^2/dx^2 P(n-2,x) + (5+2*x)*P(n-1,x) + 5*x*d/dx P(n-1,x) + x^2*d^2/dx^2 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[-(4 + 2*x + x^2)*P[n - 2, x] - (5*x + 4*x^2)*D[P[n - 2, x], x] - (x^2 + x^3)*D[P[n - 2, x], {x, 2}] + (5 + 2*x)*P[n - 1, x] + 5*x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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