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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":"ec3b893c489448004833bd6775dcd48163467908205e3f55e25110fd1e69ed64","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-be3f02dc425ece52","role":"sequence","name":"basis_map_A007318 of Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:          1;\n   2:          5,          2;\n   3:         31,         24,          4;\n   4:        247,        262,         84,          8;\n   5:       2433,       3136,       1400,        256,        16;\n   6:      28653,      42338,      23320,       6000,       720,       32;\n   7:     393279,     644216,     411916,     132160,     22480,     1920,       64;\n   8:    6169455,   10966086,    7853244,    2952152,    631120,    76832,     4928,    128;\n   9:  108934785,  207047232,  162260976,   68854016,  17369184,  2673664,   245504,  12288,   256;\n  10: 2138679765, 4300717698, 3629624880, 1696731616, 486153696, 88799424, 10359552, 744960, 29952, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(4*n^2-12*n+9+(8*n-12)*x+4*x^2)*P(n-2,x) + (4*n-4+4*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 5 + 2*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 5 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-(4*n^2 - 12*n + 9 + (8*n - 12)*x + 4*x^2)*P[n - 2, x] + (4*n - 4 + 4*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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