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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":"5d808516cacae7a994f132b4fb98244e7f6e4b4ba7ea09da9a8e6732f9131d20","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-3706be5862de5f25","role":"sequence","name":"reversal of Triangular array:  row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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:   4,    1;\n  2:  10,    9,     1;\n  3:  25,   46,    15,      1;\n  4:  43,  178,   123,     22,      1;\n  5:  64,  543,   718,    259,     30,      1;\n  6:  88, 1249,  3231,   2123,    475,     39,     1;\n  7: 115, 2422, 11334,  13253,   5180,    795,    49,    1;\n  8: 145, 4206, 31390,  65006,  42795,  11090,  1246,   60,  1;\n  9: 178, 6763, 73398, 253452, 278786, 116913, 21588, 1858, 72, 1;\n","recurrence_sentence":"The row polynomials satisfy (n+1)*P(n,x) = -(n^3+n^2-2*n)*x^2*P(n-4,x) - ((2*n^2+4*n)*x+(2*n^2+3*n)*x^2)*P(n-3,x) - (n+2-(2*n^2+2*n-2)*x+(n+1)*x^2)*P(n-2,x) + (2*n+3+(2*n+2)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 4 + x, P(2,x) = 10 + 9*x + x^2, P(3,x) = 25 + 46*x + 15*x^2 + x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + x;\nP[2, x_] := 10 + 9*x + x^2;\nP[3, x_] := 25 + 46*x + 15*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + n^2 - 2*n)*x^2*P[n - 4, x] - ((2*n^2 + 4*n)*x + (2*n^2 + 3*n)*x^2)*P[n - 3, x] - (n + 2 - (2*n^2 + 2*n - 2)*x + (n + 1)*x^2)*P[n - 2, x] + (2*n + 3 + (2*n + 2)*x)*P[n - 1, x])/(n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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