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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":"c3ef9beebafc5540575b7dcc09c83b9917ca133728a1491637e4081c3f7f7c18","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-f22372ef3b33093f","role":"sequence","name":"basis_map_A013610 of Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).","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:       0;\n  1:       3;\n  2:      30,       18;\n  3:     207,      270,       81;\n  4:    1230,     2538,     1620,      324;\n  5:    6756,    19260,    19278,     8100,     1215;\n  6:   35370,   129222,   178200,   116640,    36450,     4374;\n  7:  179301,   800550,  1414665,  1271700,   616005,   153090,    15309;\n  8:  888150,  4690890, 10141200, 11667240,  7739550,  2969946,   612360,   52488;\n  9: 4323465, 26381700, 67571010, 95085900, 80601885, 42252840, 13410684, 2361960, 177147;\n","recurrence_sentence":"The row polynomials satisfy (6*n-6)*P(n,x) = (15*n+30+(9*n+18)*x)*P(n-1,x) + (13+30*x+9*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[((15*n + 30 + (9*n + 18)*x)*P[n - 1, x] + (13 + 30*x + 9*x^2)*D[P[n - 1, x], x])/(6*n - 6)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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