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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":"14bcc0e6280eb367e883197cbc6cfbd9672dee28a0aa8f952f4fcf52d425d72d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-d938c1937a93c4df","role":"sequence","name":"Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows (n >= 1, 1 <= k <= n).","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:    2;\n   3:    6,     12;\n   4:   14,     72,      72;\n   5:   30,    300,     720,      480;\n   6:   62,   1080,    4680,     7200,     3600;\n   7:  126,   3612,   25200,    67200,    75600,     30240;\n   8:  254,  11592,  122472,   504000,   957600,    846720,    282240;\n   9:  510,  36300,  559440,  3336480,  9525600,  13970880,  10160640,   2903040;\n  10: 1022, 111960, 2455560, 20412000, 82177200, 177811200, 211680000, 130636800, 32659200;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(2+6*x+6*x^2)*P(n-2,x) - (x+3*x^2+2*x^3)*d/dx P(n-2,x) + (3+6*x)*P(n-1,x) + (x+x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 2.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[-(2 + 6*x + 6*x^2)*P[n - 2, x] - (x + 3*x^2 + 2*x^3)*D[P[n - 2, x], x] + (3 + 6*x)*P[n - 1, x] + (x + x^2)*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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