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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":"bdbe1f189952b7fd89749fd6b65500ba5f01a7e25e0f62101ba6641c34140fc4","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-dbf182f3ecfeda2a","role":"sequence","name":"Triangle of coefficients of cos(x)^n in polynomial for cos(nx).","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-658066d03ea11056","kind":"oeis_match","node_a_id":"n-0b1520d85356534b","node_b_id":null,"fingerprint":null,"oeis_id":"A110331","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":0,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A110331","oeis_nonn":false},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A110331","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-0b1520d85356534b","oeis_id":"A110331","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":14,"oeis_terms_agreeing":50,"oeis_terms_available":50,"term_count":200,"raw_size":1944}],"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:   1;\n  2:   2,    -2,   -1;\n  3:   4,    -6,   -3;\n  4:   8,   -16,   -8,    4,   1;\n  5:  16,   -40,  -20,   20,   5;\n  6:  32,   -96,  -48,   72,  18,   -6,   -1;\n  7:  64,  -224, -112,  224,  56,  -42,   -7;\n  8: 128,  -512, -256,  640, 160, -192,  -32,  8, 1;\n  9: 256, -1152, -576, 1728, 432, -720, -120, 72, 9;\n","recurrence_sentence":"The row polynomials satisfy -(2*n^2-5*n+3-(n^3-4*n^2+4*n-1)*x-(2*n^2-6*n+4)*x^2-(n-1)*x^3)*P(n,x) = -(4*n^2-10*n+6-(2*n^3-5*n^2+n+2)*x-(6*n^2-15*n+9)*x^2+(n^3-3*n^2-n+3)*x^3+(2*n^2-4*n+2)*x^4+(n-1)*x^5)*P(n-1,x) + ((2*n-2)*x-(n^2-n-1)*x^2-(4*n-3)*x^3+(n^2-n-2)*x^4+(2*n-1)*x^5+x^6)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2 - 10*n + 6 - (2*n^3 - 5*n^2 + n + 2)*x - (6*n^2 - 15*n + 9)*x^2 + (n^3 - 3*n^2 - n + 3)*x^3 + (2*n^2 - 4*n + 2)*x^4 + (n - 1)*x^5)*P[n - 1, x] + ((2*n - 2)*x - (n^2 - n - 1)*x^2 - (4*n - 3)*x^3 + (n^2 - n - 2)*x^4 + (2*n - 1)*x^5 + x^6)*D[P[n - 1, x], x])/(-2*n^2 + 5*n - 3 + (n^3 - 4*n^2 + 4*n - 1)*x + (2*n^2 - 6*n + 4)*x^2 + (n - 1)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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