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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":"4dac93cf3c3b9a284cff9065d46ec4476e6b5e1971d86c4f8e0e71a88985fe81","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:    1;\n   2:    3,      2;\n   3:    7,     18,       6;\n   4:   15,     86,     108,       24;\n   5:   31,    330,     870,      720,      120;\n   6:   63,   1142,    5220,     8760,     5400,       720;\n   7:  127,   3738,   27006,    75600,    92400,     45360,      5040;\n   8:  255,  11846,  128268,   544824,  1083600,   1038240,    423360,     40320;\n   9:  511,  36810,  577590,  3522960, 10359720,  15876000,  12489120,   4354560,   362880;\n  10: 1023, 112982, 2511540, 21230520, 87280200, 194246640, 241315200, 160876800, 48988800, 3628800;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(2+5*x+3*x^2-2*x^3)*P(n-2,x) - (x+2*x^2-x^4)*d/dx P(n-2,x) + (3+5*x-x^2)*P(n-1,x) + (x+x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 2*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 3 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-(2 + 5*x + 3*x^2 - 2*x^3)*P[n - 2, x] - (x + 2*x^2 - x^4)*D[P[n - 2, x], x] + (3 + 5*x - x^2)*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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