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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":"e2f05f5da9a02336f8591fa57b91443a2a92367a495567868dc17a1f585a5093","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-9c5c208df8605708","role":"sequence","name":"Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","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:           1;\n  2:          10,           2;\n  3:         131,          42,          3;\n  4:        2196,         892,        108,         4;\n  5:       45189,       21340,       3390,       220,        5;\n  6:     1105182,      581198,     109980,      9580,      390,       6;\n  7:    31354119,    17902710,    3841257,    405300,    22505,     630,     7;\n  8:  1012861224,   617692248,  145933032,  17685976,  1204280,   46536,   952,    8;\n  9: 36717532425, 23644951128, 6034194492, 813586536, 64257270, 3073896, 87612, 1368, 9;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(16*n^2-40*n+25+(8*n-10)*x+x^2)*P(n-2,x) + (8*n-6+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[-(16*n^2 - 40*n + 25 + (8*n - 10)*x + x^2)*P[n - 2, x] + (8*n - 6 + 2*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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