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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":"225c3544a7a0aa3530500a9d5280215bce42af9c4fe461c97c5a2b9902fbe76f","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-58f68e861f34b78a","role":"sequence","name":"Triangle read by rows, S_3(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:      0;\n   2:      1;\n   3:      7,        2;\n   4:     39,       30,        3;\n   5:    203,      318,       78,        4;\n   6:   1031,     2950,     1335,      160,       5;\n   7:   5187,    25662,    19110,     4020,     285,      6;\n   8:  25999,   215670,   248703,    81760,    9870,    462,     7;\n   9: 130123,  1777358,  3059238,  1476244,  269990,  21084,   700,    8;\n  10: 650871, 14479110, 36317655, 24746400, 6435135, 745668, 40698, 1008, 9;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(10+4*x+x^2)*P(n-2,x) - (6*x+3*x^2)*d/dx P(n-2,x) + (7+2*x)*P(n-1,x) + 3*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[-(10 + 4*x + x^2)*P[n - 2, x] - (6*x + 3*x^2)*D[P[n - 2, x], x] + (7 + 2*x)*P[n - 1, x] + 3*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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