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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":"777857f81944d64fceed03802c4a3eda012eae3c3b88feceff19f97a828c8aae","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-5939aa09e8ea2fc4","role":"sequence","name":"Coefficient triangle of certain polynomials N(5; m,x).","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:   6;\n  2:  14,   42;\n  3:  24,  168,   168;\n  4:  36,  432,  1008,    504;\n  5:  50,  900,  3600,   4200,    1260;\n  6:  66, 1650,  9900,  19800,   13860,    2772;\n  7:  84, 2772, 23100,  69300,   83160,   38808,   5544;\n  8: 104, 4368, 48048, 200200,  360360,  288288,  96096,  10296;\n  9: 126, 6552, 91728, 504504, 1261260, 1513512, 864864, 216216, 18018;\n","recurrence_sentence":"The row polynomials satisfy (n^2+3*n-4)*P(n,x) = (n^2+5*n+(3*n^2+9*n+12)*x)*P(n-1,x) + ((2*n+4)*x-(2*n+4)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 6.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 6;\nP[n_, x_] := P[n, x] = Expand[Together[((n^2 + 5*n + (3*n^2 + 9*n + 12)*x)*P[n - 1, x] + ((2*n + 4)*x - (2*n + 4)*x^2)*D[P[n - 1, x], x])/(n^2 + 3*n - 4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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