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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":"72280e2d6c5d283a80d91258fd77da84613dd5f3cb0efea0d60bde7a017f58ef","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-735e311cbe9a718e","role":"sequence","name":"Triangle T(n, k) read by rows: T(n, k) = (Sum_{i=0..k} (-1)^i * binomial(k, i) * (n+i)^n) * (-1)^k / k! for 0 <= 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:         0;\n  1:         1;\n  2:         5,         2;\n  3:        37,        24,         3;\n  4:       369,       302,        66,        4;\n  5:      4651,      4380,      1275,      140,        5;\n  6:     70993,     73502,     25650,     3860,      255,      6;\n  7:   1273609,   1412208,    559503,   103320,     9520,    420,     7;\n  8:  26269505,  30683774,  13360914,  2856084,   327810,  20412,   644,   8;\n  9: 612579511, 745368180, 349258395, 83414100, 11227335, 879984, 39522, 936, 9;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2-n+(2*n-2)*x+x^2)*P(n-2,x) - (n^2-n+(3*n-3)*x+2*x^2)*d/dx P(n-2,x) - ((n-1)*x+x^2)*d^2/dx^2 P(n-2,x) + (2*n+1+2*x)*P(n-1,x) + (n+1+2*x)*d/dx P(n-1,x) + x*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[-(n^2 - n + (2*n - 2)*x + x^2)*P[n - 2, x] - (n^2 - n + (3*n - 3)*x + 2*x^2)*D[P[n - 2, x], x] - ((n - 1)*x + x^2)*D[P[n - 2, x], {x, 2}] + (2*n + 1 + 2*x)*P[n - 1, x] + (n + 1 + 2*x)*D[P[n - 1, x], x] + x*D[P[n - 1, x], {x, 2}]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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