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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":"5f3fde4f91e4aeef9ceb7e4cdec16b0d07bbd01def547ab33a373faa690df240","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-7d6d6baad7692ddc","role":"sequence","name":"Triangle read by rows, S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; 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:        1;\n   2:        4,         1;\n   3:       19,        12,         1;\n   4:      106,       121,        24,         1;\n   5:      661,      1200,       418,        40,        1;\n   6:     4384,     12121,      6540,      1070,       60,       1;\n   7:    29959,    125292,     98251,     24320,     2285,      84,      1;\n   8:   207526,   1320481,   1454964,    510111,    71190,    4319,    112,     1;\n   9:  1446121,  14116800,  21472228,  10249680,  1989246,  176736,   7476,   144,   1;\n  10: 10103164, 152412241, 317184360, 201082540, 52098480, 6348342, 388920, 12108, 180, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(21-6*x-3*x^2)*P(n-2,x) - (12*x-12*x^2)*d/dx P(n-2,x) + (10-2*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 4 + x;\nP[n_, x_] := P[n, x] = Expand[-(21 - 6*x - 3*x^2)*P[n - 2, x] - (12*x - 12*x^2)*D[P[n - 2, x], x] + (10 - 2*x)*P[n - 1, x] + 4*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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