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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":"0d5607aa599c61daf90f5e26da274b86c2776500cb5224af6e1e7af483c07944","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-d3c3090977105807","role":"sequence","name":"Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k=0 and 0<=k<=2n.","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: 5, 6, 1;\n 2: 9, 44, 42, 12, 1;\n 3: 13, 114, 315, 292, 111, 18, 1;\n 4: 17, 216, 1012, 2120, 1974, 904, 212, 24, 1;\n 5: 21, 350, 2325, 7800, 13890, 13044, 6810, 2040, 345, 30, 1;\n 6: 25, 516, 4446, 20660, 55935, 89640, 84804, 48744, 17415, 3860, 510, 36, 1;\n 7: 29, 714, 7567, 45052, 165081, 384566, 572691, 544968, 336903, 137494, 37149, 6524, 707, 42, 1;\n 8: 33, 944, 11880, 86352, 400316, 1234800, 2573592, 3631760, 3472038, 2271504, 1028440, 323568, 70140, 10192, 936, 48, 1;\n 9: 37, 1206, 17577, 150960, 847764, 3267432, 8837556, 16907472, 22898646, 21974596, 15039198, 7398864, 2627268, 670824, 121284, 15024, 1197, 54, 1;\n","recurrence_sentence":"The row polynomials satisfy (4*n-3+(2*n+2)*x+x^2)*P(n,x) = (4*n+1+(18*n+8)*x+(12*n+18)*x^2+(2*n+8)*x^3+x^4)*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[((4*n + 1 + (18*n + 8)*x + (12*n + 18)*x^2 + (2*n + 8)*x^3 + x^4)*P[n - 1, x])/(4*n - 3 + (2*n + 2)*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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