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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":"165bde34e3dbf58df6458f92b6bfa95fa68dfe0697cd54bff7a65434987d7e41","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-a6f3b35778192ffb","role":"sequence","name":"Triangle T(n,k) = number of k-part order-consecutive partitions of n (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:  0;\n  1:  1;\n  2:  3,   2;\n  3:  6,  12,    3;\n  4: 10,  38,   30,     4;\n  5: 15,  90,  135,    60,     5;\n  6: 21, 180,  423,   360,   105,    6;\n  7: 28, 322, 1071,  1428,   805,  168,    7;\n  8: 36, 532, 2352,  4428,  3920, 1596,  252,   8;\n  9: 45, 828, 4662, 11628, 14535, 9324, 2898, 360, 9;\n","recurrence_sentence":"The row polynomials satisfy (n^2-n-(5*n^2-17*n+12)*x+(4*n^2-10*n+6)*x^2)*P(n,x) = (n^2+n-(5*n^2-15*n)*x+(n^2+3*n)*x^2+(12*n^2-22*n)*x^3)*P(n-1,x) + (2*n*x-6*n*x^2-6*n*x^3-8*n*x^4)*d/dx 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[Together[((n^2 + n - (5*n^2 - 15*n)*x + (n^2 + 3*n)*x^2 + (12*n^2 - 22*n)*x^3)*P[n - 1, x] + (2*n*x - 6*n*x^2 - 6*n*x^3 - 8*n*x^4)*D[P[n - 1, x], x])/(n^2 - n - (5*n^2 - 17*n + 12)*x + (4*n^2 - 10*n + 6)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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