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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":"dbf09641813bc3d932e0b1b158e0dd330c26a5680e86156f8cb90823ad35af50","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-7466465df6cec0d5","role":"sequence","name":"Triangle T(n,k) read by rows: T(n, k) = (m*n - m*k + 1)*T(n - 1, k - 1) + (5*k - 4)*(m*k - (m - 1))*T(n - 1, k) where m = 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:       7,        2;\n   4:      43,       36,         3;\n   5:     259,      482,       102,         4;\n   6:    1555,     5820,      2355,       220,        5;\n   7:    9331,    67130,     46410,      7760,      405,       6;\n   8:   55987,   757092,    843255,    224840,    20230,     672,     7;\n   9:  335923,  8439986,  14627718,   5845980,   807030,   45108,  1036,    8;\n  10: 2015539, 93511692, 246703467, 142269204, 28290255, 2366784, 89922, 1512, 9;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(6+2*x+x^2)*P(n-2,x) - (5*x+5*x^2)*d/dx P(n-2,x) + (7+2*x)*P(n-1,x) + 5*x*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[-(6 + 2*x + x^2)*P[n - 2, x] - (5*x + 5*x^2)*D[P[n - 2, x], x] + (7 + 2*x)*P[n - 1, x] + 5*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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