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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":"ce0f641e8a71480d89315125a14d37781a39d1b3c8ab21c69157eebd94fa166c","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-849de7332df28657","role":"sequence","name":"Triangle T(n, k) read by rows: T(n, k) = Sum_{j=0..n} binomial(j, n-k) * |Stirling1(n, n-j)|.","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:     1,      1;\n  2:     1,      5,       2;\n  3:     2,     16,      25,        6;\n  4:     6,     64,     167,      142,       24;\n  5:    24,    316,    1133,     1633,      926,      120;\n  6:   120,   1868,    8437,    16937,    16384,     6876,      720;\n  7:   720,  12888,   69816,   178328,   242091,   173836,    57492,     5040;\n  8:  5040, 101736,  640488,  1976776,  3435429,  3478092,  1970828,   535536,   40320;\n  9: 40320, 904608, 6474660, 23355020, 49278057, 64126641, 51389416, 23938524, 5505264, 362880;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(n^3-4*n^2+4*n+(2*n^3-6*n^2+4*n)*x+(n^3-2*n^2+n)*x^2)*P(n-2,x) + (2*n^2-4*n+1+(2*n^2-2*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 4*n^2 + 4*n + (2*n^3 - 6*n^2 + 4*n)*x + (n^3 - 2*n^2 + n)*x^2)*P[n - 2, x] + (2*n^2 - 4*n + 1 + (2*n^2 - 2*n)*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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