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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":"02198958d6651f4cdd2713a25ba2a62bc395c06641234b200019112e246569ae","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-4d7e6ff489a3ffc4","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k peaks (1 <= k <= n).","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:       2;\n   3:      16,        8;\n   4:     102,      120,       24;\n   5:     608,     1168,      576,       64;\n   6:    3530,     9520,     7920,     2240,      160;\n   7:   20256,    70680,    84480,    41280,     7680,      384;\n   8:  115598,   496104,   780360,   557760,   182560,    24192,     896;\n   9:  658048,  3356192,  6577536,  6272896,  3046400,   720384,   71680,   2048;\n  10: 3742290, 22128480, 52063200, 62536320, 40985280, 14547456, 2612736, 202752, 4608;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-2*n-4)*P(n,x) = (10*n^2-10*n+4+(12*n^2-28*n+8)*x)*P(n-1,x) + (2*n-1-(8*n-4)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 2.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((10*n^2 - 10*n + 4 + (12*n^2 - 28*n + 8)*x)*P[n - 1, x] + (2*n - 1 - (8*n - 4)*x^2)*D[P[n - 1, x], x])/(2*n^2 - 2*n - 4)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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