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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":"dda68e302f17b72724e14743d89fdfe6c36a7a31fb641c7fd75240d3eac7ef9f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-4479b0484a11b442","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of ternary trees with n edges and having k middle edges (n >= 0, k >= 0).","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:      0;\n   2:      1;\n   3:      6,      2;\n   4:     28,     24,       3;\n   5:    120,    180,      60,       4;\n   6:    495,   1100,     660,     120,      5;\n   7:   2002,   6006,    5460,    1820,    210,      6;\n   8:   8008,  30576,   38220,   19600,   4200,    336,     7;\n   9:  31824, 148512,  239904,  171360,  57120,   8568,   504,   8;\n  10: 125970, 697680, 1395360, 1302336, 610470, 143640, 15960, 720, 9;\n","recurrence_sentence":"The row polynomials satisfy (4*n^3-6*n^2-4*n)*P(n,x) = (15*n^3-22*n^2+17*n-6+(12*n^3-32*n^2+16*n)*x)*P(n-1,x) + (n^2-3*n+2+(7*n^2-3*n-2)*x-(8*n^2-6*n)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((15*n^3 - 22*n^2 + 17*n - 6 + (12*n^3 - 32*n^2 + 16*n)*x)*P[n - 1, x] + (n^2 - 3*n + 2 + (7*n^2 - 3*n - 2)*x - (8*n^2 - 6*n)*x^2)*D[P[n - 1, x], x])/(4*n^3 - 6*n^2 - 4*n)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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