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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":"b2c46e17600f2becfda4ee14e621005dae0d53891e0409198b46b0ca32244b7c","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-d55b2cfe05a04774","role":"sequence","name":"Triangle of B-analogs of Stirling numbers of the second kind.","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:    4,      2;\n   4:   13,     18,       3;\n   5:   40,    116,      48,       4;\n   6:  121,    660,     510,     100,       5;\n   7:  364,   3542,    4560,    1580,     180,      6;\n   8: 1093,  18438,   37233,   20300,    3955,    294,     7;\n   9: 3280,  94376,  288288,  232344,   68880,   8568,   448,   8;\n  10: 9841, 478440, 2159580, 2475480, 1048110, 194040, 16716, 648, 9;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(3+2*x+x^2)*P(n-2,x) - (2*x+2*x^2)*d/dx P(n-2,x) + (4+2*x)*P(n-1,x) + 2*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[-(3 + 2*x + x^2)*P[n - 2, x] - (2*x + 2*x^2)*D[P[n - 2, x], x] + (4 + 2*x)*P[n - 1, x] + 2*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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