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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":"e14b9c7f2e4483f75f0cde94bde2a50ed13063b58671bc83dec5253c256e8089","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-2c11b65ec5b65118","role":"sequence","name":"reversal of Triangle read by rows: T(n,k) = k*(k+1)*binomial(n,k)/2 (1 <= k <= n).","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:   1;\n  1:   5,    2;\n  2:  15,   15,     3;\n  3:  34,   60,    30,     4;\n  4:  65,  170,   150,    50,     5;\n  5: 111,  390,   510,   300,    75,     6;\n  6: 175,  777,  1365,  1190,   525,   105,    7;\n  7: 260, 1400,  3108,  3640,  2380,   840,  140,    8;\n  8: 369, 2340,  6300,  9324,  8190,  4284, 1260,  180,   9;\n  9: 505, 3690, 11700, 21000, 23310, 16380, 7140, 1800, 225, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(1+4*x+6*x^2+4*x^3+x^4)*P(n-4,x) + (4+12*x+12*x^2+4*x^3)*P(n-3,x) - (6+12*x+6*x^2)*P(n-2,x) + (4+4*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 5 + 2*x, P(2,x) = 15 + 15*x + 3*x^2, P(3,x) = 34 + 60*x + 30*x^2 + 4*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + 2*x;\nP[2, x_] := 15 + 15*x + 3*x^2;\nP[3, x_] := 34 + 60*x + 30*x^2 + 4*x^3;\nP[n_, x_] := P[n, x] = Expand[-(1 + 4*x + 6*x^2 + 4*x^3 + x^4)*P[n - 4, x] + (4 + 12*x + 12*x^2 + 4*x^3)*P[n - 3, x] - (6 + 12*x + 6*x^2)*P[n - 2, x] + (4 + 4*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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