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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":"f939c01fedd2710d7275e4422210dc2229410bb3090b5c62e939a458ae091792","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-8ac1223942f7b8c5","role":"sequence","name":"reversal of Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).","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,   30;\n  3: 12,   92,   168;\n  4: 16,  186,   696,    836;\n  5: 20,  312,  1776,   4364,   3900;\n  6: 24,  470,  3600,  13464,  24560,   17466;\n  7: 28,  660,  6360,  32040,  88920,  128808,   76048;\n  8: 32,  882, 10248,  65020, 243120,  535428,  642768,  324360;\n  9: 36, 1136, 15456, 118356, 557620, 1654344, 3016944, 3090984, 1361844;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2-6*n+2+(6*n^2-18*n+12)*x)*P(n,x) = (4*n^2-2*n+(30*n^2-37*n-1)*x+(60*n^2-132*n+24)*x^2+(36*n^2-108*n+72)*x^3)*P(n-1,x) - (2*n*x+(19*n-3)*x^2+(48*n-24)*x^3+(36*n-36)*x^4)*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[((4*n^2 - 2*n + (30*n^2 - 37*n - 1)*x + (60*n^2 - 132*n + 24)*x^2 + (36*n^2 - 108*n + 72)*x^3)*P[n - 1, x] - (2*n*x + (19*n - 3)*x^2 + (48*n - 24)*x^3 + (36*n - 36)*x^4)*D[P[n - 1, x], x])/(4*n^2 - 6*n + 2 + (6*n^2 - 18*n + 12)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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