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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":"2b57ede0cc3997f402c2df80cb0b49b6add9124742710147c3df348930ebab4b","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-4c2a0f8cb422ddaa","role":"sequence","name":"Sheffer triangle S2[4,1] = (exp(x), exp(4*x) - 1).","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;\n  2:      24,        32;\n  3:     124,       480,        192;\n  4:     624,      5312,       5376,        1024;\n  5:    3124,     52800,     101760,       46080,        5120;\n  6:   15624,    500192,    1639680,     1326080,      337920,      24576;\n  7:   78124,   4626720,   24316992,    31288320,    13726720,    2236416,    114688;\n  8:  390624,  42265472,  343881216,   661592064,   444702720,  121798656,  13762560,   524288;\n  9: 1953124, 383514240, 4724048640, 13081098240, 12646717440, 5179539456, 967507968, 80216064, 2359296;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(5+8*x+16*x^2)*P(n-2,x) - (4*x+16*x^2)*d/dx P(n-2,x) + (6+8*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[-(5 + 8*x + 16*x^2)*P[n - 2, x] - (4*x + 16*x^2)*D[P[n - 2, x], x] + (6 + 8*x)*P[n - 1, x] + 4*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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