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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":"e4580bbe595cff8262a910e6b9ea3e85046750c6ffe2d5e5fc70f9652ec01d8c","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-c15181c2ebe13c1d","role":"sequence","name":"Triangle of numbers a(n,k), 0 <= k <= n: number of set partitions of {1,2,...,n} in which exactly k of the blocks have been distinguished.","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:    10,     12,      3;\n  4:    37,     62,     30,      4;\n  5:   151,    320,    225,     60,      5;\n  6:   674,   1712,   1560,    620,    105,     6;\n  7:  3263,   9604,  10668,   5600,   1435,   168,    7;\n  8: 17007,  56674,  73878,  47964,  16450,  2940,  252,   8;\n  9: 94828, 351792, 524415, 404124, 173355, 41832, 5502, 360, 9;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(1+2*x+x^2)*P(n-2,x) - (1+2*x+x^2)*d/dx P(n-2,x) + (3+2*x)*P(n-1,x) + (1+x)*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[-(1 + 2*x + x^2)*P[n - 2, x] - (1 + 2*x + x^2)*D[P[n - 2, x], x] + (3 + 2*x)*P[n - 1, x] + (1 + x)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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