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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":"9faf65e4cdc82594cc915f88b1c36362592605710959bfb81325bedee3c6858c","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-5b4c394836f6a454","role":"sequence","name":"theta_plus_one of Riordan array (1, 3+3x).","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: 1;\n  1: 6,   6;\n  2: 6,  60,  27;\n  3: 0, 108, 378,  108;\n  4: 0,  54, 999, 1944,   405;\n  5: 0,   0, 972, 6804,  8910,   1458;\n  6: 0,   0, 324, 9828, 38880,  37908,    5103;\n  7: 0,   0,   0, 6480, 74520, 198288,  153090,   17496;\n  8: 0,   0,   0, 1620, 73305, 473850,  933849,  594864,   59049;\n  9: 0,   0,   0,    0, 36450, 619650, 2673972, 4146552, 2243862, 196830;\n","recurrence_sentence":"The row polynomials satisfy (n^3-n^2-2*n^2*x+(3*n^2-8*n)*x^2+3*n*x^3)*P(n,x) = -(2*n^3-8*n^2+6*n-(16*n^2-16*n-12)*x+(6*n^2+8*n+28)*x^2-(12*n-24)*x^3-18*x^4)*P(n-1,x) + ((4*n^2-4*n)*x+(3*n^2-11*n-8)*x^2+(6*n-14)*x^3+(9*n+6)*x^4+9*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 - 8*n^2 + 6*n - (16*n^2 - 16*n - 12)*x + (6*n^2 + 8*n + 28)*x^2 - (12*n - 24)*x^3 - 18*x^4)*P[n - 1, x] + ((4*n^2 - 4*n)*x + (3*n^2 - 11*n - 8)*x^2 + (6*n - 14)*x^3 + (9*n + 6)*x^4 + 9*x^5)*D[P[n - 1, x], x])/(n^3 - n^2 - 2*n^2*x + (3*n^2 - 8*n)*x^2 + 3*n*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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