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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":"6d2216f926eeb107fe03ecab780614e2f7da26959ce043da5aa9240f5ad404f2","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: 0;\n 1: 4, 2;\n 2: 8, 36, 24, 4;\n 3: 12, 102, 264, 204, 60, 6;\n 4: 16, 200, 912, 1816, 1520, 600, 112, 8;\n 5: 20, 330, 2160, 7080, 12120, 10620, 5040, 1320, 180, 10;\n 6: 24, 492, 4200, 19260, 51120, 79416, 71568, 38520, 12600, 2460, 264, 12;\n 7: 28, 686, 7224, 42644, 154420, 353682, 513744, 471576, 277956, 106610, 26488, 4116, 364, 14;\n 8: 32, 912, 11424, 82544, 379680, 1158864, 2380448, 3291696, 3060576, 1931440, 835296, 247632, 49504, 6384, 480, 16;\n 9: 36, 1170, 16992, 145296, 811440, 3105144, 8320032, 15718896, 20933784, 19648620, 13074336, 6210288, 2109744, 508536, 84960, 9360, 612, 18;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (n+4*n*x+n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4 + 2*x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 4 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((n + 4*n*x + n*x^2)*P[n - 1, x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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