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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":"6cd43dfe52009c077f2987cc528812b5c8cfca426b7171be5f84d63d99685bc7","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-cab14672969f6507","role":"sequence","name":"Galton triangle T(n, k) = T(n-1, k-1) + (3k-2)*T(n-1, k) read by rows.","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:     2,       1;\n   3:     6,       7,        1;\n   4:    22,      45,       15,        1;\n   5:    86,     295,      171,       26,       1;\n   6:   342,    1981,     1795,      465,      40,       1;\n   7:  1366,   13527,    18291,     7190,    1035,      57,      1;\n   8:  5462,   93325,   184275,   103621,   22190,    2016,     77,    1;\n   9: 21846,  647815,  1848211,  1437786,  430241,   57302,   3570,  100,   1;\n  10: 87382, 4512861, 18503955, 19515945, 7858620, 1439487, 129990, 5886, 126, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(4-x-2*x^2)*P(n-2,x) - (3*x-6*x^2)*d/dx P(n-2,x) + (5-x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[-(4 - x - 2*x^2)*P[n - 2, x] - (3*x - 6*x^2)*D[P[n - 2, x], x] + (5 - x)*P[n - 1, x] + 3*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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